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1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | 1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | how many armies did the germans send to attack the americans in the battle of the bulge ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | still , there was n't a surrender from the japanese , and so , let me write this . this is august 6th , and so on august 9th , you have another atomic bomb dropped on nagasaki . at the same time that this is happening , you have the soviet union invading manchuria . | i know there was the 6th panzer army trying to capture the bridges over the meuse river , and the 5th panzer army encircling bastogne , but was there another german army involved ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | is it split up ? what influence will the different allied victors have in the different countries of world war ii ? so this is happening in yalta in february of 1945 . | why were certain countries allied and what is the benefit of it ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | 1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | how was the battle of the bulge start in germany ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | so once again , these cities are really being destroyed . this is all-out war . but still the japanese have not surrendered . | how could hitler have won the war ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | so the allies have won in europe , but everything is not over yet . world war ii is still going on in the pacific . so let 's go down to the pacific theater . | was the atom bomb used only with the intent of ending the war in pacific ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | this is all-out war . but still the japanese have not surrendered . so in april , actually april going into june , you have to remember april in europe , this is when it was pretty much done with . | even after the huge destruction of japanese troops and armaments in the philippines ' battles , how was japan still able to fight and think of defending themselves ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | you have the soviets who are now getting involved in the pacific , the americans seem to be able to literally destroy japanese cities at will , and so on august 14 , 1945 , you have the japanese surrender . japanese surrender , and then this does not become truly official until september 2 , 1945 , which is called vj d... | but why is it victory in japan day when japan lost ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | by january of 1945 , they 're able to take warsaw , and from there , they continue to march westward towards berlin . now the writing is on the wall at this point . the allies look like they are going to win , and so they meet at yalta , the major powers , to discuss what happens to europe after world war ii . | where did the phrase writing on the wall come from ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | all the while this is happening , even though the writing is on the wall that the allies are going to win the war , they were n't taking anything for granted . they wanted to absolutely force a surrender by the axis powers in germany in particular . so they continue to firebomb major cities of germany , and firebombing... | if the us had n't entered the war would the axis powers have won ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | this is august 6th , and so on august 9th , you have another atomic bomb dropped on nagasaki . at the same time that this is happening , you have the soviet union invading manchuria . you have the soviet union invading manchuria , and so enough was enough . you have the soviets who are now getting involved in the pacif... | what was hitler 's plan if he had capitulated the soviet union ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | this is august 6th , and so on august 9th , you have another atomic bomb dropped on nagasaki . at the same time that this is happening , you have the soviet union invading manchuria . you have the soviet union invading manchuria , and so enough was enough . you have the soviets who are now getting involved in the pacif... | can you say what percentage , approximately , of german army was killed by soviet union and allies armies ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | in parallel to all of this , the allies , especially the u.s. , has had the manhattan project to develop nuclear weapons , and they decide to use them , truman is now president , in order to accelerate the end of the war . and so in august 6th , they drop an atomic bomb on hiroshima , essentially destroying that city .... | how big can an atomic explosion be ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | japanese surrender , and then this does not become truly official until september 2 , 1945 , which is called vj day , victory in japan day . that marks the real end of world war ii . obviously , world war ii ended at the end of april , may , officially early may in europe , but now in early september it is officially e... | how advanced were the germans compared to the allies during world war two ? |
1945 really represents the final throes of world war ii . in europe , you might remember , at the end of 1944 , we have the battle of the bulge starting , which is an incredibly bloody battle . the u.s. forces , which kind of take the brunt on the allied side , lose roughly 20,000 troops , but by january , they 're ab... | several other major leaders , nazi leaders , commit suicide as we go into this period right over here . essentially the war is won in europe at this point , and that 's made official on may 8th , may 8th , which is a good day to remember , known as ve day , victory in europe , victory in europe day . so the allies have... | why was n't the m26 pershing medium tank deployed in large numbers in europe after development was finished ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | and s is going to be all of the values equivalent to that or greater than for g equal to 0 . if s is equal to 0 , g is greater than or equal to 15 . so if s is equal to 0 , g is greater than or equal to 15 . | why are we able to make g equal to 0 ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so if g is 0 , s is going to be greater than or equal to 15 . let me put it this way . so if i 'm going to graph this one right here . | why did you put g as zero ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph . and that 's why we have some graph paper over here . | is it possible to use substitution and elimination to solve inequalities ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | just a little over 28 . so 28.08 . so that is , g is 0 , s is 28 . | this is really nit-picky , but is n't 25/0.89 rounded to 28.09 not 28.08 ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so the overlap is going to be this region right here . below the orange graph and above the blue graph , including both of them . so if you pick any combination -- so if he buys 4 games and 14 songs , that would work . | how do you represent only integer solutions on the graph ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | 0.89s plus 1.99g is less than or equal to 25 . this is a starting point . let 's just draw the line 0.89s plus 1.99 is equal to 25 . | does the point ( 9,6 ) of `` intersection '' of the two `` lines '' have any significance ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so the overlap is going to be this region right here . below the orange graph and above the blue graph , including both of them . so if you pick any combination -- so if he buys 4 games and 14 songs , that would work . | how can we differentiate between numer of units and prices on the graph ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so just the positive quadrant right here . let me draw the axes . so let 's make the vertical axis that i 'm drawing right here , let 's make that the vertical axis and let 's call that the song axis . | would it not be easier to convert both equations to y ( = ) mx+b and draw the lines ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph . and that 's why we have some graph paper over here . | what is the system of linear combinations ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so if g is 0 , s is going to be greater than or equal to 15 . let me put it this way . so if i 'm going to graph this one right here . | is there an easy way to remember how to figure out the inequalities and solve the solutions ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so the area that satisfies this second constraint is everything below this graph . now we want the region that satisfies both constraints . so it 's going to be the overlap of the regions that satisfy one of the two . | what are the constraints on x and y ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so it would look something like this . this is always the hardest part . let me see how well i can connect these two dots . | is n't the unwanted part supposed to be shaded instead of the part that satisfies the constraints ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so if s is equal to 0 , g is greater than or equal to 15 . so g is greater than or equal to 15 . so the boundary line , s plus g is equal to 15 , we would just have to connect these two dots . | why did sal say that s+g is greater than or equal to 15 ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so let 's make the vertical axis that i 'm drawing right here , let 's make that the vertical axis and let 's call that the song axis . so that 's the number of songs he buys . let me make sure you can see that . | why does sal write `` songs '' right next to s= # when you already know that s= # is the number of songs ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . | can someone please tell me what is linear programming used for in practical life ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | let 's just draw the line 0.89s plus 1.99 is equal to 25 . and then we could think about what region the less than would represent . oh , 1.99g . | how can you explain what graphs of equations and inequalities represent ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | let me make sure you can see that . that is the song axis . and then let 's make this , this horizontal , that 's going to be the number of games he buys . | is it incorrect to switch the song axis and game axis ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | so this would be 4 , 8 , 12 , 16 , 20 , so on and so forth . and this would be 4 -- this obviously would be 0 -- 4 , 8 , 12 , 16 , 20 , and so on . so let 's see if we can graph these two constraints . | what would the optimal solution be ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | let me see . i should get a line tool for this . so that 's pretty good . | how do you know to `` shade '' one side of the line as apposed to the other side ? |
luis receives a gift card worth $ 25 to an online retailer that sells digital music and games . each song costs $ 0.89 and each game costs $ 1.99 . he wants to buy at least 15 items with this card . set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph ... | let me try my best to connect these dots . so it would look something like this . this is always the hardest part . | like when you divide both sides you have to flip the sign ( < , > ) ? |
the function f is defined over the real numbers . this table gives select values of f. we have our table here for these x values . it gives the corresponding f of x . what is a reasonable estimate for the limit of f of x as x approaches one from the left ? so pause this video and see if you can figure it out on your o... | so a reasonable estimate for the limit as x approaches one from the left of f of x , it looks like x , it looks like f of x right over here is approaching two . we do n't know for sure . that 's why they 're saying what is a reasonable estimate . | how to know that the function is unbounded ? |
( smooth piano music ) - [ steve ] we 're in the louvre , in paris , looking at one of their most famous objects . this is the law code stele of king hammurabi . - [ beth ] it 's interesting to me that this is one of the most popular objects to look at here and i think it 's because of our modern interest and reliance ... | ( smooth piano music ) - [ steve ] we 're in the louvre , in paris , looking at one of their most famous objects . this is the law code stele of king hammurabi . - [ beth ] it 's interesting to me that this is one of the most popular objects to look at here and i think it 's because of our modern interest and reliance ... | how literate was the average person when this law code was written ? |
( smooth piano music ) - [ steve ] we 're in the louvre , in paris , looking at one of their most famous objects . this is the law code stele of king hammurabi . - [ beth ] it 's interesting to me that this is one of the most popular objects to look at here and i think it 's because of our modern interest and reliance ... | he 's giving to the king a scepter and a ring , these are signs of power . - hammurabi is demonstrating here that these are divine laws . - [ steve ] that his authority comes from shamash . | would it have given some people an advantage over others because only some could read the laws ? |
hey everyone . so in the last video i introduced this thing called the second partial derivative test , and if you have some kind of multivariable function or really just a two variable function is what this applies to , something that 's f of x , y and it outputs a number . when you 're looking for places where it ha... | it 's kind of like there 's disagreement in different directions over whether it should be a maximum or a minimum . and if h equals zero , the test is n't good enough . you would have to do something else to figure it out . | is there any reason why the expression used in the test is the determinant of the hessian matrix ? |
hey everyone . so in the last video i introduced this thing called the second partial derivative test , and if you have some kind of multivariable function or really just a two variable function is what this applies to , something that 's f of x , y and it outputs a number . when you 're looking for places where it ha... | now when you find a point like this , in order to test whether it 's a local maximum or a local minimum or a saddle point without actually looking at the graph , 'cause you do n't always have the ability to do that at your disposal , the first step is to compute this long value , and this is the thing i wan na give int... | will it be a saddle point if the second partial derivative of x and the second partial derivative of y have different signs ? |
hey everyone . so in the last video i introduced this thing called the second partial derivative test , and if you have some kind of multivariable function or really just a two variable function is what this applies to , something that 's f of x , y and it outputs a number . when you 're looking for places where it ha... | but still , looking at only three different terms to take into account possible disagreement in infinitely many directions actually feels like quite the surprise . and if you want the full , rigorous justification for why this is the case , why this second partial derivative test works and kind of , an airtight argumen... | where is the airtight argument ? |
: so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | now , we can also talk about large-scale mutations , which instead of being at the level of individual nucleotides , are usually seen at the chromosomal level and can affect many genes , instead of just a few base pairs . so , first we 'll talk about translocation , which is when a gene from one chromosome is swapped f... | i was taught in class that inversion refers to the event when a gene segment breaks away and rejoins at the same spot but in the reversed orientation ( so basically the gene is `` inverted '' ) which definition is correct ? |
: so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | now , it 's important to see that translocation refers to gene swapping between nonhomologous chromosomes , which means that if this blue chromosome were chromosome 10 , then the green one could be any chromosome aside from chromosome 10 , and this is what sets translocation apart from the process of crossing over that... | are transposons an example of an inversion ( large-scale mutation ) , or are they two completely different entities ? |
: so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | so , first we 'll talk about translocation , which is when a gene from one chromosome is swapped for another gene on a different chromosome . now , it 's important to see that translocation refers to gene swapping between nonhomologous chromosomes , which means that if this blue chromosome were chromosome 10 , then the... | how many homologous chromosomes does a human have ? |
: so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | so , what did we learn ? well , first we learned that small-scale mutations affect the dna at the nucleotide level , and of these small-scale mutations , we have point mutations , which can be caused by transitions , transversions , and mispairings , and we also have frame-shift mutations , which can be caused by inser... | are point mutations the same thing as base pair mutations ? |
: so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | this would then be transcribed into an rna strand with repeating gag units and an extra g on the end , and our three codons would be the three gag units , which would then each translate to a glutamate amino acid . now , one way you can cause a frame-shift mutation is through an insertion , and that 's when an extra dn... | during an insertion frame shift mutation , what happens to the extra residue on the dna strand ? |
: so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | a transversion is when either adenine or guanine is swapped for either cytosine or thymine , and in this type of base substitution , you have either a purine being replaced with a pyrimidine or a pyrimidine being replaced with a purine . now , the last kind of mutation that can lead to a point mutation is a mispairing ... | how does mismatching ( mispairing ) differ from transition and transversion ? |
: so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | : so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | can someone explain 3 ( ' ) prime and 5 ( ' ) prime ? |
: so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | : so , today we 're going to talk about the causes of genetic mutations , but first let 's just do a quick review of the idea that mutations are mistakes in a cell 's dna , and there are two main types of mutations that we see when we look at a cell 's dna , and the first is called point mutations , and that 's when o... | why are they added , and when do they switch between dna and rna because the nucleotides are changing as well ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | now , when we have this infection , there are going to be some local immune cells that your body has that 's going to be fighting these bacteria and trying to kill them . but your body also has more powerful , really , defenses , which consist of b cells and t cells . and these guys are part of the adaptive immune syst... | once the b cells and t cells detect a threat , how do they find the place of infection ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and that 's what this structure is called . it 's called a lymph node . it 's called a node because it 's generally a smallish object in your body , and it looks somewhat circular . | is it true that lymph node massage can have an effect on your health ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and it 's also convenient , because what leads to it is the lymphatic vessel . so these bacteria and some of these macrophages carrying the bacteria will get swept along . and they 'll be in contact with these b and t cells , who can then multiply and specialize and get ready to fight this infection over here . | what if the bacteria is in the cell and not in the interstitial fluid ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | the infection usually stays in your urinary tract . or maybe you get pneumonia . the infection stays in your lungs . | do they get extracted into lymph nodes or how do they get killed ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and it 's also convenient , because what leads to it is the lymphatic vessel . so these bacteria and some of these macrophages carrying the bacteria will get swept along . and they 'll be in contact with these b and t cells , who can then multiply and specialize and get ready to fight this infection over here . | how do bacteria ( unicellular organism ) enter ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and it 's also convenient , because what leads to it is the lymphatic vessel . so these bacteria and some of these macrophages carrying the bacteria will get swept along . and they 'll be in contact with these b and t cells , who can then multiply and specialize and get ready to fight this infection over here . | how do the macrophages get into the interstitial fluid in the first place ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . so let 's start with places where you might actually feel lymph nodes on... | are there any known conditions where the person is born with fewer lymphatic vessels ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and that 's what this structure is called . it 's called a lymph node . it 's called a node because it 's generally a smallish object in your body , and it looks somewhat circular . | does any lymph node go to the heart directly ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | how come the infection will go into the lymphatic system but not a blood vessel ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and by that , i mean that they usually stay in one place . so think about when you get an infected cut . usually -- let 's say , it 's on your finger -- the infection will stay on your finger . | so you could think of the b and t cells as `` police '' for our immune system when we get infected with a type of bacteria or virus ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | the infection stays in your lungs . so if the infection was actually in your blood , it would immediately go all over the body . and that does happen sometimes . | an infection in the blood would be like sepsis right ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . so let 's start with places where you might actually feel lymph nodes on your own body . so often , you can feel the lymph nodes that are in th... | how can we feel the lymph nodes in the body ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | let 's actually give this guy a neck to be generous . so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . | so the lymph nodes act as filters , but what happens to the `` waste '' that is filtered ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | let 's actually give this guy a neck to be generous . so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . | can we assume that these lymph nodes are 100 % efficient ( in most cases ) at 'filtering ' bacteria out of the lymph fluid before it reenters the bloodstream ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | it 's going to continue , and it 's going to eventually carry fluid and such back into the blood . so ultimately , the fluid is going to end up back in the blood . and so actually , another nice side effect of having these lymph nodes here interspersed throughout the lymphatic vessels is that they 'll actually filter a... | what happens to the bacteria in the infection that the lymph nodes filter out before putting the fluid back into the blood ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and that 's what this structure is called . it 's called a lymph node . it 's called a node because it 's generally a smallish object in your body , and it looks somewhat circular . | what happens if the lymph node does not properly kill a particular bacteria or bacterium and it then enters the bloodstream ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | let 's actually give this guy a neck to be generous . so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . | if the lymph nodes are doing there job how is it possible for someone to get bacteria in there blood stream ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and so in this way , you have about 600 of these lymph nodes sprinkled about in your body . and the cool fact to remember is that any lymph that 's coming out of any tissue in your body will pass through at least one lymph node before it goes back to your blood . so even if it 's out in your little pinky finger or some... | also is it possible that lymph will pass through more than 1 lymph node before going back to the heart ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | so all those are along lymph vessels that are coming down . and then the lymph vessels that are coming here from your arms -- those will have lymph nodes along them . and so in this way , you have about 600 of these lymph nodes sprinkled about in your body . and the cool fact to remember is that any lymph that 's comin... | if the lymph are damaged or infected will the bacteria enter the blood stream and affect the whole body ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | and that 's what this structure is called . it 's called a lymph node . it 's called a node because it 's generally a smallish object in your body , and it looks somewhat circular . | what happens if you injure a lymph node ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | let 's actually give this guy a neck to be generous . so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . | i 'm slightly confused about the contents of lymph nodes - as b and t cells are wbcs , how did they get to the lymph or lymph node ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | let 's actually give this guy a neck to be generous . so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . | why do lymph nodees clog and what happens when they are not cleared ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | let 's actually give this guy a neck to be generous . so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . | how much lymph is reintroduced into the circulation each day ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | let 's actually give this guy a neck to be generous . so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . | how do macrophages migrate tissue to the local lymph nodes ? |
we 've talked about how the purpose of the lymphatic system is to collect fluid that 's squeezed out of blood vessels like this one -- to collect that fluid and eventually bring it back into the blood so that we do n't lose too much fluid . so this is a blood vessel . around it , we have some cells that are maybe being... | let 's actually give this guy a neck to be generous . so the lymph nodes are , obviously , located all along lymph vessels . but because there are 600 of them , they 're located along the larger lymphatic vessels that have already collected many of the little lymphatic vessels and consolidated them . | why do lymph nodes swell and become painful when there is an infection ? |
what we want to do is find the derivative of this g of x and at first it can look intimidating . we have a sine of x here . we have a cosine of x . we have this crazy expression here with a pi over cube root of x we 're squaring the whole thing and at first it might seem intimidating . but as we 'll see in this video ... | using our existing derivative properties using what we know about the power rule which tells us the derivative with respect to x . of x to the n is equal to n times x to the n minus one , we 've see that multiple times . we also need to use the fact that the derivative of cosine of x is equal to negative the sine of x ... | why does n't the last term become 2pi*-2/3x^-5/3 ? |
what we want to do is find the derivative of this g of x and at first it can look intimidating . we have a sine of x here . we have a cosine of x . we have this crazy expression here with a pi over cube root of x we 're squaring the whole thing and at first it might seem intimidating . but as we 'll see in this video ... | but as we 'll see in this video , we can actually do this with the tools already in our tool kit . using our existing derivative properties using what we know about the power rule which tells us the derivative with respect to x . of x to the n is equal to n times x to the n minus one , we 've see that multiple times . | how would you compute the derivative of -sinx ? |
what we want to do is find the derivative of this g of x and at first it can look intimidating . we have a sine of x here . we have a cosine of x . we have this crazy expression here with a pi over cube root of x we 're squaring the whole thing and at first it might seem intimidating . but as we 'll see in this video ... | and the derivative of cosine of x so it 's minus three times the derivative of cosine of x is negative sine of x . negative sine of x . and then finally here in the yellow we just apply the power rule . | do you factor the negative sign so that it 's the derivative of - ( sinx ) which is -cosx ? |
what we want to do is find the derivative of this g of x and at first it can look intimidating . we have a sine of x here . we have a cosine of x . we have this crazy expression here with a pi over cube root of x we 're squaring the whole thing and at first it might seem intimidating . but as we 'll see in this video ... | so you have the negative two thirds . you multiply the exponent times the coefficient . it might look confusing , pi squared , but that 's just a number . | why do we need to multiply the exponent to the rest of the terms ? |
what we want to do is find the derivative of this g of x and at first it can look intimidating . we have a sine of x here . we have a cosine of x . we have this crazy expression here with a pi over cube root of x we 're squaring the whole thing and at first it might seem intimidating . but as we 'll see in this video ... | negative sine of x . and then finally here in the yellow we just apply the power rule . so , we have the negative two thirds , actually , let 's not forget this minus sign i 'm gon na write it out here . | why cant we just apply the power rule straight away ? |
what we want to do is find the derivative of this g of x and at first it can look intimidating . we have a sine of x here . we have a cosine of x . we have this crazy expression here with a pi over cube root of x we 're squaring the whole thing and at first it might seem intimidating . but as we 'll see in this video ... | so , so let me just do this on the side here . so , pi pi over the cube root of x squared . well that 's the same thing . | ( pi/cube root of x ) ^2 to 2 ( pi/cube root of x ) ^1 ? |
what we want to do is find the derivative of this g of x and at first it can look intimidating . we have a sine of x here . we have a cosine of x . we have this crazy expression here with a pi over cube root of x we 're squaring the whole thing and at first it might seem intimidating . but as we 'll see in this video ... | so it 's gon na be negative and then you have negative two thirds times pi squared . times pi squared . times x to the negative two thirds minus one power . | pi squared is a constant value , so when differentiated its value should be zero right ? |
what we want to do is find the derivative of this g of x and at first it can look intimidating . we have a sine of x here . we have a cosine of x . we have this crazy expression here with a pi over cube root of x we 're squaring the whole thing and at first it might seem intimidating . but as we 'll see in this video ... | using our existing derivative properties using what we know about the power rule which tells us the derivative with respect to x . of x to the n is equal to n times x to the n minus one , we 've see that multiple times . we also need to use the fact that the derivative of cosine of x is equal to negative the sine of x ... | why do n't we differentiate pie and x individually ? |
dr. steven zucker : there 's nothing subtle about 17th century dutch genre painting . so often , we 're shown interactions that are wonderfully bawdy and wonderfully explicit . there is an exception , however . jan vermeer 's paintings often are riddles . they give us suggestions of narratives . dr. beth harris : and ... | he stands beside a table with a beautiful carpet on it , and he has his hand on a jug of wine . he looks like he 's ready to refill the young woman 's glass . she 's got it up to her mouth , and she 's just finishing it off . | is there any significance in how she is holding the glass -- from the base ? |
dr. steven zucker : there 's nothing subtle about 17th century dutch genre painting . so often , we 're shown interactions that are wonderfully bawdy and wonderfully explicit . there is an exception , however . jan vermeer 's paintings often are riddles . they give us suggestions of narratives . dr. beth harris : and ... | it 's about alignment and things being askew . dr. beth harris : and that 's also symbolized in the musical instrument , which is used in vermeer 's paintings to suggest both harmony and frivolity . so which way is this going to go ? | is the musical instrument a guitar ? |
dr. steven zucker : there 's nothing subtle about 17th century dutch genre painting . so often , we 're shown interactions that are wonderfully bawdy and wonderfully explicit . there is an exception , however . jan vermeer 's paintings often are riddles . they give us suggestions of narratives . dr. beth harris : and ... | dr. steven zucker : there 's nothing subtle about 17th century dutch genre painting . so often , we 're shown interactions that are wonderfully bawdy and wonderfully explicit . | how come renaissance era artwork never had this much depth and realism to it ? |
dr. steven zucker : there 's nothing subtle about 17th century dutch genre painting . so often , we 're shown interactions that are wonderfully bawdy and wonderfully explicit . there is an exception , however . jan vermeer 's paintings often are riddles . they give us suggestions of narratives . dr. beth harris : and ... | dr. steven zucker : there 's nothing subtle about 17th century dutch genre painting . so often , we 're shown interactions that are wonderfully bawdy and wonderfully explicit . | do you know what the values might be ? |
dr. steven zucker : there 's nothing subtle about 17th century dutch genre painting . so often , we 're shown interactions that are wonderfully bawdy and wonderfully explicit . there is an exception , however . jan vermeer 's paintings often are riddles . they give us suggestions of narratives . dr. beth harris : and ... | then you 've got a series of folds in the drapery that catch the light and sort of expand as they move down towards his arm . and that motion is picked up by the beautiful gold brocade in the woman 's dress and then the folds on her hip . and so there really is a kind of harmony between those figures . | is the girls dress silk by the shine on the dress ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | this is equal to 7 times the cube root of 10 . and this is about as simplified as we can get just using hand arithmetic . if you want to get the exact number here , you 're probably best off using a calculator . | is sal writing out the problems on the black space by hand or with an actual computer pointer ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . | how did sal get rid of the cube root and the ^3 ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . | are there other ways ( other than prime factorization ) to find the cube root of a number ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | and the cube root of 10 , we just leave it as 10 . we know the prime factorization of 10 is 2 times 5 , so you 're not going to just get a very simple integer answer here . you would get some decimal answer here , but here you get a very clear integer answer . | * how do we know that the final surd form is the simplest ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | and from our exponent properties , we know that this is the exact same thing as the cube root of 2 times 5 times the cube root -- so let me do that in that same , just so we see what colors we 're dealing with . so the cube root of 2 times 5 , which is the cube root of 10 , times the cube root -- and i think you see wh... | how do you solve the cube root of a number such as 2 or 7 ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | keeping track of the colors is the hard part . and the cube root of 10 , we just leave it as 10 . we know the prime factorization of 10 is 2 times 5 , so you 're not going to just get a very simple integer answer here . | how can we calculate cube root of 10 ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | how do we know which number to put outside or inside ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . | is there another way besides prime factorization to find out cube roots ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | so our entire thing simplifies . this is equal to 7 times the cube root of 10 . and this is about as simplified as we can get just using hand arithmetic . | how we calculate the cubic root of 10 multiply 7 to get that result 343 i mean how to do reversed solution ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | and from our exponent properties , we know that this is the exact same thing as the cube root of 2 times 5 times the cube root -- so let me do that in that same , just so we see what colors we 're dealing with . so the cube root of 2 times 5 , which is the cube root of 10 , times the cube root -- and i think you see wh... | what if it is a cube root of a negative non-perfect cube ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | so if i have 1,715 , and i 'm going to divide it by 5 . 5 does n't go into 1 . it goes into 17 three times . | where would the negative sign go ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | and 49 might jump out at you . it can be factored into 7 times 7 . so this is interesting . | for example , in this problem would the negative sign be on the 7 or 10 ? |
let 's see if we can find the cube root of 3,430 . and if you 're like me , it does n't jump out of your mind what number times that same number times that same number -- if you have three of those numbers and you were to multiply them together -- would be equal to 3,430 . so what i 'm going to do is to try to prime fa... | and from our exponent properties , we know that this is the exact same thing as the cube root of 2 times 5 times the cube root -- so let me do that in that same , just so we see what colors we 're dealing with . so the cube root of 2 times 5 , which is the cube root of 10 , times the cube root -- and i think you see wh... | is there ever a remainder when you are doing the cube root or square root ? |
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