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( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . - ( laughing ) what is color ? color is a sensation . ( chiming ) - color is a very personal thing . color is very subjective . it can trigger like a visceral response . l... | ( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . | what does the world look like for color blind people ? |
( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . - ( laughing ) what is color ? color is a sensation . ( chiming ) - color is a very personal thing . color is very subjective . it can trigger like a visceral response . l... | ( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . | in other words , what do color blind people see ? |
( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . - ( laughing ) what is color ? color is a sensation . ( chiming ) - color is a very personal thing . color is very subjective . it can trigger like a visceral response . l... | ( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . | how does color convey a character 's mood ? |
( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . - ( laughing ) what is color ? color is a sensation . ( chiming ) - color is a very personal thing . color is very subjective . it can trigger like a visceral response . l... | light comprises different wavelengths of energy , and when that energy comes through our pupil onto our retina inside our eye , it becomes nerve impulses , signals , and eventually gets processed by the brain . so the light is out in the real world . the color only really exists inside your brain when you perceive it . | what does it mean 'so the light is out in the real world ' ? |
( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . - ( laughing ) what is color ? color is a sensation . ( chiming ) - color is a very personal thing . color is very subjective . it can trigger like a visceral response . l... | ( chiming music ) - what do i think color is ? ( laughing ) what is color ? ( gentle cheerful music ) - ( laughing ) i ca n't answer that question . | so , as final , what is color ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . | which word phrase could correspond to the variable expressions n/8 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | 15 times 15 is 225 . so 1.5 times 1.5 is 2.25 . so 1.5 squared is 2.25 . | is there a difference between ( 5^2 ) & ( 5 ) ^2 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | and 2.25 times 6 -- so let me just multiply that out . 2.25 times 6 . let 's see . | thus my equation was a^2 + b^2 ( 6 ) my question is , will doing it this way cause me problems later on down the road ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | so it 's 1.5 by 1.5 by 1.5 . what is the total surface area that she has to paint ? well , we know that the surface area of each cube is going to be 6x squared , where x is the dimensions of that cube . | what does sal mean when he uses the term surface area ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | 15 times 15 is 225 . so 1.5 times 1.5 is 2.25 . so 1.5 squared is 2.25 . | someone can explain me why 1.5 + 1.5 is 3 , but 1.5 square is 2.25 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . | why do you leave the exponent inside of the parenthese rather than outside of it ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | 6 times 2 is 12 , plus 1 is 13 . i have two numbers behind the decimal -- 13.5 . so it 's going to be 13.5 . | how do you differentiate between the multiplication dot and the decimal point ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | one cube has side length 2 . so this is one cube right over here . i 'll do my best to draw it . | a carat is usually used to indicate a variable , right so you would use the method to solve a problem ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | and 2.25 times 6 -- so let me just multiply that out . 2.25 times 6 . let 's see . | is there a difference between 6 ( 2^2 ) and 6 ( 2 ) ^2 and 6*2^2 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | 15 times 15 is 225 . so 1.5 times 1.5 is 2.25 . so 1.5 squared is 2.25 . | can someone explain how 1.5x1.5= 2.25 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | 15 times 15 is 225 . so 1.5 times 1.5 is 2.25 . so 1.5 squared is 2.25 . | 1x1=1 and .5x.5=0.25 together they should be 1.25 not 2.25 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | jolene has two cube-shaped containers that she wants to paint . one cube has side length 2 . so this is one cube right over here . | why do we have to add a 'square ' after the variable ( or the length ) ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | so it 's going to be 6x squared . jolene has two cube-shaped containers that she wants to paint . one cube has side length 2 . so this is one cube right over here . i 'll do my best to draw it . | do we cube the number if we are calculating volume ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | 15 times 15 is 225 . so 1.5 times 1.5 is 2.25 . so 1.5 squared is 2.25 . | why did sal square 2 and 1.5 instead of putting them to the third power ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | so we 're just going to add these two things . and so if we were to compute this first one right over here , this is going to be 6 times 4 . this is 24 . | if vlad 's experiment starts with n=8100 molecules , how many molecules will be left after t=4 minutes ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | so 1.5 times 1.5 is 2.25 . so 1.5 squared is 2.25 . and 2.25 times 6 -- so let me just multiply that out . | do you have to multiply a variable by the squared sign or vy the constant ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | so it 's 1.5 by 1.5 by 1.5 . what is the total surface area that she has to paint ? well , we know that the surface area of each cube is going to be 6x squared , where x is the dimensions of that cube . so the surface area of this cube right over here is going to be 6 . | could the surface area of the cube also be written as x^3 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | so it 's 1.5 by 1.5 by 1.5 . what is the total surface area that she has to paint ? well , we know that the surface area of each cube is going to be 6x squared , where x is the dimensions of that cube . so the surface area of this cube right over here is going to be 6 . and now -- let me do it in that color of that cub... | why is the surface area of the second cube 13.25 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | it has side length 1.5 . so it 's 1.5 by 1.5 by 1.5 . what is the total surface area that she has to paint ? | am pacific time , what does it mean to evaluate an expression ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . | is the voice for sal really his voice ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | so it 's 1.5 by 1.5 by 1.5 . what is the total surface area that she has to paint ? well , we know that the surface area of each cube is going to be 6x squared , where x is the dimensions of that cube . | what is a surface area ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . | what does geometry have to do with algebra variables and algebra problems ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | jolene has two cube-shaped containers that she wants to paint . one cube has side length 2 . so this is one cube right over here . | does the 6xsquared equation work for all word problems that give you the side of a cube ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | and 2.25 times 6 -- so let me just multiply that out . 2.25 times 6 . let 's see . | is 6x^2 always true for cubes ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | and 2.25 times 6 -- so let me just multiply that out . 2.25 times 6 . let 's see . | for example what does 6x^2 mean ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . | but why does the exponent go inside the parentheses ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | and 2.25 times 6 -- so let me just multiply that out . 2.25 times 6 . let 's see . | then would the exponent apply to the `` 6*2 '' or just the 2 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . | are variables used in higher levels of math ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . | why do the sun and moon travel across the sky ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | and 2.25 times 6 -- so let me just multiply that out . 2.25 times 6 . let 's see . | what is the difference between ( 2^2 ) and ( 2 ) ^2 ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . | does it matter when the exponent is in or outside the brackets ? |
the surface area of a cube is equal to the sum of the areas of its six sides . let 's just visualize that . i like to visualize things . so if that 's the cube , we can see three sides . three sides are facing us . but then if it was transparent , we see that there are actually six sides of a cube . so there 's this on... | jolene has two cube-shaped containers that she wants to paint . one cube has side length 2 . so this is one cube right over here . | i do n't get why `` the surface area of a cube with the side length x is given the expression 6x^2 '' ... why is x raised to the power of 2 ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | we 're compounding by 5 % every time . we 're increasing by a factor of 1.05 . or another way of thinking about it , by a factor of 105 % every week . | why is 5 % = 1.05 ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | we 're compounding by 5 % every time . we 're increasing by a factor of 1.05 . or another way of thinking about it , by a factor of 105 % every week . | how in the world does 5 % .=1.05 ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | this can be described by a linear model . the number of wild hogs in arkansas increases by a factor of 3 every 5 years . so a factor of 3 every 5 years . | how would you write the problem that states `` the number of wild dogs in arkansas increases by a factor of 3 every 5 years '' as an equation ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | we 're increasing by 5 % . increasing by 5 % means you 're 1.05 times as big as you were before increasing . so it 's really this function is exponential because w increases by a factor of 1.05 each time t increases by 1 . that , right over there , is the right answer . | how did sal know to increase the function by 1.05 ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . | are linear and exponential models more important than other functions ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | : a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . | does a polynomial in x vary arithmetically , or geometrically , with x ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | they 're saying by 5 % . after one week it 'll be 1.05 times 40 kilograms . after another week it 'll be 1.05 times that , it 'll be 5 percent more . | is the equation y=40 ( 1.05 ) ^i here i is the week , correct ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . | can a linear function be expressed into an exponential function ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . | so can we rule that multiplication/division is always exponential and addition/subtraction is always linear ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | so this is definitely exponential . if it was increasing $ 10 per year , then it would be linear . but here we 're increasing by a percentage . | how would you write the first question as an equation ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | if we grow by 10 % , that 's increasing by a factor of 110 % or 1.1 . so this is definitely exponential . if it was increasing $ 10 per year , then it would be linear . | overall general question about exponential functions : how do you know when they will be horizontal or vertical exponential functions ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | so let 's see which if these choices describe that . this function is linear , no , we do n't have to even read that . this function is linear , nope . | i do n't understand is n't the term exponetial have to do with powers of a number ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | if we grow by 10 % , that 's increasing by a factor of 110 % or 1.1 . so this is definitely exponential . if it was increasing $ 10 per year , then it would be linear . | is the exponential similar to compound interest ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | this can be described by a linear model . the number of wild hogs in arkansas increases by a factor of 3 every 5 years . so a factor of 3 every 5 years . | what equation would describe the number of wild hogs ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | they 're saying by 5 % . after one week it 'll be 1.05 times 40 kilograms . after another week it 'll be 1.05 times that , it 'll be 5 percent more . | how come for the first cow question , you increase by 1.05 times instead of 0.05 ? |
: a newborn calf weighs 40 kilograms . each week its weight increases by 5 % . let w be the weight in kilograms of the calf after t weeks . is w a linear function or an exponential function ? so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so ... | so , if w were a linear function , that means that every week that goes by , the weight would increase by the same amount . so let 's say that every week that went by , the weight increases ... or , really , they 're talking about mass here . the mass increased by 5 kilograms . | is n't kilogram a measure of mass rather than weight ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | why is n't 0 to the 0th power 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | what 's the difference between 1^0 and 1^1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | why is anything to the zero power equal to 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | now let 's try some other interesting scenarios . let 's start try some negative numbers . so let 's take negative 1 . | for all negative numbers : odd = negative even= positive ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | at 1.13 why does something to the 0th power = 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and then you could see that if you went to negative 1 to the fourth power . negative 1 the fourth power ? well , you could start with a 1 and then multiply it by negative 1 four times , so a negative 1 times negative 1 , times negative 1 , times negative 1 , which is just going to be equal to positive 1 . | what is a power honestly ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so if someone were to ask you -- we already established that if someone were to take 1 to the , i do n't know , 1 millionth power , this is just going to be equal to 1 . if someone told you let 's take negative 1 and raise it to the 1 millionth power , well , 1 million is an even number , so this is still going to be e... | how is taking the -1 power of a nonzero number equivalent to taking the reciprocal of that number ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | if every exponent begins by multiplying by 1 , as a rule , then are n't all 1^x expressions short by 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so let 's take negative 1 . and let 's first raise it to the 0 power . so once again , this is just going , based on this definition , this is starting with a 1 and then multiplying it by this number 0 times . | what is the answer to 0 to the power of 0 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | in the video , it talks about 1 to the any real number is 1 , but what if the exponent is imaginary ( for instance , i , which is the square root of negative 1 , or the trans-infinite numbers ) ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . then you multiply it by negative 1 , you 're going to get positive 1 . then you multiply it by negative 1 again to get negative 1 . | would -1 raised to the power of infinity be negative or positive ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | if i take a number say 8^0 then , we will do this 1*8*0 =1*0 =0 right ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | why is it 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | so , ( -1 ) to the 1,000,000 power = -1. correct ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so let 's take negative 1 . and let 's first raise it to the 0 power . so once again , this is just going , based on this definition , this is starting with a 1 and then multiplying it by this number 0 times . | so 0 to the 3rd power would be one ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | what would the answer be for `` 1 to the half or 0.5 power '' ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | what would be the answer for `` 1 to the 1/2 power '' ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | 2 minutes 20 seconds in , how can ( -1 ) to the zero power be 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and just as a little bit of intuition here , you could literally view this as our other definition of exponentiation , which is you start with a 1 , and this number says how many times you 're going to multiply that 1 times this number . so 1 times 1 zero times is just going to be 1 . and that was a little bit clearer ... | is n't 1 to the 0th power supposed to be zero , since it means 1 was multiplied 0 times , therefore no 1 's ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so negative 1 to the second power -- well , we could start with a 1 . we could start with a 1 , and then multiply it by negative 1 two times -- multiply it by negative 1 twice . and what 's this going to be equal to ? | why do you have to multiply the exponent by 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | now let 's try some other interesting scenarios . let 's start try some negative numbers . so let 's take negative 1 . | also , do i always have to put parentheses for negative numbers ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | if someone told you let 's take negative 1 and raise it to the 1 millionth power , well , 1 million is an even number , so this is still going to be equal to positive 1 . but if you took negative 1 to the 999,999th power , this is an odd number . so this is going to be equal to negative 1 . | 2.why does negative number become positive number when it is multiplied by another negative number ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so let 's take negative 1 . and let 's first raise it to the 0 power . so once again , this is just going , based on this definition , this is starting with a 1 and then multiplying it by this number 0 times . well , that means we 're just not going to multiply it by this number . | i was thinking , the definition of exponenet i think was basically multiplying the base by itself by the number of times the power displays , so how come 5^0 is one , when it was never multiplied against itself even once , should n't it be zero ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . then you multiply it by negative 1 , you 're going to get positive 1 . | how can that be possible when we are squaring a negative number ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and just as a little bit of intuition here , you could literally view this as our other definition of exponentiation , which is you start with a 1 , and this number says how many times you 're going to multiply that 1 times this number . so 1 times 1 zero times is just going to be 1 . and that was a little bit clearer ... | how come when -1 was multiplied by zero , it became +1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | why does the -1 become positive 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | so for negative one , basically if the exponent is odd , it will be -1 , and if it is a even exponent is will be 1 , right ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . | how do you type a fraction on a key board ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | if 56^ ( a ) = 5.6^ ( b ) = 10^ ( c ) , show that ( 1 ) / ( a ) = ( 1 ) / ( b ) + ( 1 ) / ( c ) what is needs to be done on this equation ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . | can you identify patterns while solving all exponents ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | if someone told you let 's take negative 1 and raise it to the 1 millionth power , well , 1 million is an even number , so this is still going to be equal to positive 1 . but if you took negative 1 to the 999,999th power , this is an odd number . so this is going to be equal to negative 1 . | so if any number to the power of 0 is one then what about a negative number , would the answer still be positive one ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . then you multiply it by negative 1 , you 're going to get positive 1 . | is there a negative number exponent ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | how do i describe how to write any power of 1 without multiplying ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . then you multiply it by negative 1 , you 're going to get positive 1 . | what do you do when there is a negative number for an example : -1 to the -4th power ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and if you take it to an even power , you 're going to get 1 because a negative times a negative is going to be the positive . and you 're going to have an even number of negatives , so that you 're always going to have negative times negatives . so this right over here , this is even . even is going to be positive 1 .... | 0 what is even number ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you have one , two , three , four , five , six , seven , eight 1 's , and then you 're going to multiply them together . and if you were to do that , you would get well , 1 times 1 is 1 , times 1 -- it does n't matter how many times you multiply 1 by 1 . you are going to just get 1 . | would their be able to have 1 even exponent number x1 that will equal ( -1 ) ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | does the -1^999,999 work with other negative numbers such as -2 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | does anyone know why when i do negative 1 to the power of an even number on my calculator it still shows as negative 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so once again , this is just going , based on this definition , this is starting with a 1 and then multiplying it by this number 0 times . well , that means we 're just not going to multiply it by this number . so you 're just going to get a 1 . | do we put the number in parentheses when we substitute or does it go in without them ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . then you multiply it by negative 1 , you 're going to get positive 1 . | what is 1 to the negative 1st power ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | if 1 or negative 1 to any power is 1 or -1 , than what is 1 or -1 to the 1st or -1st power ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and then you could see that if you went to negative 1 to the fourth power . negative 1 the fourth power ? well , you could start with a 1 and then multiply it by negative 1 four times , so a negative 1 times negative 1 , times negative 1 , times negative 1 , which is just going to be equal to positive 1 . | what example in real life is there of the negative power ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and then you could see that if you went to negative 1 to the fourth power . negative 1 the fourth power ? well , you could start with a 1 and then multiply it by negative 1 four times , so a negative 1 times negative 1 , times negative 1 , times negative 1 , which is just going to be equal to positive 1 . | is there a zero power ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | why does -1 to the zero power equal 1 and not -1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | and once again , by our old definition , you could also just say , hey , ignoring this one , because that 's not going to change the value , we took two negative 1 's and we 're multiplying them . well , negative 1 times negative 1 is 1 . and i think you see a pattern forming . | how do you do if the question is 1^54245 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so you see the pattern . negative 1 to the 0 power is 1 . negative 1 to the first power is negative 1 . | why does a number to the zero power equal 1 ? |
let 's think about exponents with ones and zeroes . so let 's take the number 1 , and let 's raise it to the eighth power . so we 've already seen that there 's two ways of thinking about this . you could literally view this as taking eight 1 's , and then multiplying them together . so let 's do that . so you have one... | so 1 to any power is just going to be equal to 1 . and you might say , hey , what about 1 to the 0 power ? well , we 've already said anything to 0 power , except for 0 -- that 's where we 're going to -- it 's actually up for debate . but anything to the 0 power is going to be equal to 1 . | how is 0^0 up for debate ? |
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