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we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | well , this is just going to be 5 . 5 times 5 is 25 . or you could say , 25 is equal to 5 squared . | is there an easier way to find that number that if given the certain exponent ( 5 ) makes it into the 32 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so you 're starting to see something interesting . 4 to the 1/2 is equal to 2 , 2 squared is equal to 4 . so let 's get a couple more examples of this , just so you make sure you get what 's going on . | would it be correct to say that 4^1/2 * 4^1/2 = 4 because you add the halves , which become 1 , as another way of showing 4^1/2 is equal to the square root of 4 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so the cube root of 8 , or 8 to the 1/3 , is just going to be equal to 2 . this says hey , give me the number that if i say that number , times that number , times that number , i 'm going to get 8 . well , that number is 2 because 2 to the third power is 8 . | so to be clear , when given a rational exponent the rules change to say , no longer is the number being raised a `` factor '' to be multiplied by itself by the number of times its raised by but is now the `` product '' you 're looking to arrive at by the number of times given by the denominator ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . | how can you solve a fraction witha fraction as its exponent ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | does that mean that that the number next to the square root is the amount of times in which you are suppose to find the cubed square root of ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . | how do you type the radical sign in the computer ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | but either way , this is going to result in 4 times 4 is 16 , times 4 is 64 . we also know a little bit about negative exponents . so for example , if i were take 4 to the negative 3 power , we know this negative tells us to take the reciprocal 1/4 to the third . | what if the base is negative ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so this says hey , give me the number that if i were to multiply that number , or i were to repeatedly multiply that number five times , what is that , i would get 32 . well , 32 is the same thing as 2 times 2 times 2 times 2 times 2 . so 2 is that number , that if i were to multiply it five times , then i 'm going to ... | so how is 32 the same thing as 2.2.2.2. ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | well , this is just going to be 5 . 5 times 5 is 25 . or you could say , 25 is equal to 5 squared . | how would you solve a fraction with a fractional exponent , like 94/17^2/5 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | do a few more examples of that . what is 64 to the 1/3 power ? well , we already know that 4 times 4 times 4 is 64 . | what do you do if you take 25 and raise it to a power where the numerator is n't one like 25 to the 3/2 power ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | well , we already know that 8 is equal to 2 to the third power . so the cube root of 8 , or 8 to the 1/3 , is just going to be equal to 2 . this says hey , give me the number that if i say that number , times that number , times that number , i 'm going to get 8 . | so following the logic of 8^1/3 = 2 does that make 8^2/3 = 4 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | what does principal root mean ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | so does that mean that by saying ( for example ) 4 to the power of 1/2 you are just asking the square root of four ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | how to type 6th root ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so this says hey , give me the number that if i were to multiply that number , or i were to repeatedly multiply that number five times , what is that , i would get 32 . well , 32 is the same thing as 2 times 2 times 2 times 2 times 2 . so 2 is that number , that if i were to multiply it five times , then i 'm going to ... | what if the exponent is a fraction that is like , 3^3/2 or 4^3/2 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so the cube root of 8 , or 8 to the 1/3 , is just going to be equal to 2 . this says hey , give me the number that if i say that number , times that number , times that number , i 'm going to get 8 . well , that number is 2 because 2 to the third power is 8 . | when you are square rooting a positive number , should it be positive/negative that number ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | is a fifth root possible ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | well , this is just going to be 5 . 5 times 5 is 25 . or you could say , 25 is equal to 5 squared . | if a jet can travel 600 mph , how long in seconds will it take the jet to travel 5 miles ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we already know 4 to the third is 64 , so this is going to be 1/64 . now let 's think about fractional exponents . so we 're going to think about what is 4 to the 1/2 power . | is there any general way to calculate the value of an expression having a fractional exponent ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so you 're starting to see something interesting . 4 to the 1/2 is equal to 2 , 2 squared is equal to 4 . so let 's get a couple more examples of this , just so you make sure you get what 's going on . | like here sal finds ( 32 ) ^ ( 1/5 ) as 2 x 2 x 2 x 2 x 2 but what about a number where this is not as clear such as ( 4 ) ^ ( 1/4 ) or ( 3 ) ^ ( 1/3 ) ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so this says hey , give me the number that if i were to multiply that number , or i were to repeatedly multiply that number five times , what is that , i would get 32 . well , 32 is the same thing as 2 times 2 times 2 times 2 times 2 . so 2 is that number , that if i were to multiply it five times , then i 'm going to ... | what do you do if you have an exponent like 2/3 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | i know what a root is but what is a cube root ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | or you could say , 25 is equal to 5 squared . now , let 's think about what happens when you take something to the 1/3 power . so let 's imagine taking 8 to the 1/3 power . | what happens when the fractional exponent is n't as clearly defined as something like1/2 or 1/3 as the exponent ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so this says hey , give me the number that if i were to multiply that number , or i were to repeatedly multiply that number five times , what is that , i would get 32 . well , 32 is the same thing as 2 times 2 times 2 times 2 times 2 . so 2 is that number , that if i were to multiply it five times , then i 'm going to ... | what if the exponent is 2/3 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | well , this is just going to be 5 . 5 times 5 is 25 . or you could say , 25 is equal to 5 squared . | -1/-5 x 5 + ( 7-9 ) + 11 to the 2nd power ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so you 're starting to see something interesting . 4 to the 1/2 is equal to 2 , 2 squared is equal to 4 . so let 's get a couple more examples of this , just so you make sure you get what 's going on . | why is it that 4 to the power of 1/2 is the square root of 4 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | let 's do one more like this . what is 25 to the 1/2 going to be ? well , this is just going to be 5 . | should i cancel out the 10s , m 's , and exponents to end up with 1^6 / 25 yr-1 , or do i move the tens to the end and end up with ( 1 / 25 yr-1 ) x 10^6 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | but either way , this is going to result in 4 times 4 is 16 , times 4 is 64 . we also know a little bit about negative exponents . so for example , if i were take 4 to the negative 3 power , we know this negative tells us to take the reciprocal 1/4 to the third . | how about the exponent being a negative fraction ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | this says hey , give me the number that if i say that number , times that number , times that number , i 'm going to get 8 . well , that number is 2 because 2 to the third power is 8 . do a few more examples of that . | but what about if the 1/2 power is used on an irregular square root number ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | does the mean you would find the cube root of it ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | is there such a thing as a fourth root ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . | why is algebra called algebra ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | do a few more examples of that . what is 64 to the 1/3 power ? well , we already know that 4 times 4 times 4 is 64 . so this is going to be 4 . | why is 4 to the 1/3 power = the cube root of 4 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | do a few more examples of that . what is 64 to the 1/3 power ? well , we already know that 4 times 4 times 4 is 64 . | does 'm ' to the 'a ' power with an index of 'b ' = 'm ' to the 'a'/'b ' power ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but either way , this is going to result in 4 times 4 is 16 , times 4 is 64 . we also know a little bit about negative exponents . | is there any way to calculate roots ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so the cube root of 8 , or 8 to the 1/3 , is just going to be equal to 2 . this says hey , give me the number that if i say that number , times that number , times that number , i 'm going to get 8 . well , that number is 2 because 2 to the third power is 8 . | is there a way to find out the roots of a number ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | is sal saying that by definition the fractional exponent of a number is the principal root only ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | do we always ignore the negative root ( assuming one exists ) when we use this notation ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | what is 64 to the 1/3 power ? well , we already know that 4 times 4 times 4 is 64 . so this is going to be 4 . | why is it that 4 to the power of one half is the same as the square root of 4 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . | how do you calculate the decimal ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . | what do you do when you have a variable inside a radical with a fraction as an exponent ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | the principal root of 9 , that 's just going to be equal to 3 . and likewise , we could 've also said that 3 squared is , or let me write it this way , that 9 is equal to 3 squared . these are both true statements . | how do i write a radical in simplest form ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so this says hey , give me the number that if i were to multiply that number , or i were to repeatedly multiply that number five times , what is that , i would get 32 . well , 32 is the same thing as 2 times 2 times 2 times 2 times 2 . so 2 is that number , that if i were to multiply it five times , then i 'm going to ... | 32 ^2/5 what is the value of the expression below ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | so basically a principle root is just negative and a postive of a root number ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | the principal root of 9 , that 's just going to be equal to 3 . and likewise , we could 've also said that 3 squared is , or let me write it this way , that 9 is equal to 3 squared . these are both true statements . | how do you write an expression in radical form ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so this says hey , give me the number that if i were to multiply that number , or i were to repeatedly multiply that number five times , what is that , i would get 32 . well , 32 is the same thing as 2 times 2 times 2 times 2 times 2 . so 2 is that number , that if i were to multiply it five times , then i 'm going to ... | how do you find a number^2/3 for eg ; how do you find ( 29.5 ) ^2/3 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so you 're starting to see something interesting . 4 to the 1/2 is equal to 2 , 2 squared is equal to 4 . so let 's get a couple more examples of this , just so you make sure you get what 's going on . | so would 4^1/2 be equal to only +2 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so this says hey , give me the number that if i were to multiply that number , or i were to repeatedly multiply that number five times , what is that , i would get 32 . well , 32 is the same thing as 2 times 2 times 2 times 2 times 2 . so 2 is that number , that if i were to multiply it five times , then i 'm going to ... | why is ^1/2 = ^2 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . | but what do you do when the integer ( not the exponent ) is a fraction ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so the cube root of 8 , or 8 to the 1/3 , is just going to be equal to 2 . this says hey , give me the number that if i say that number , times that number , times that number , i 'm going to get 8 . well , that number is 2 because 2 to the third power is 8 . | how would you find the fractional root of a number irrational number such as : 40^1/5 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | but either way , this is going to result in 4 times 4 is 16 , times 4 is 64 . we also know a little bit about negative exponents . so for example , if i were take 4 to the negative 3 power , we know this negative tells us to take the reciprocal 1/4 to the third . | how do you conceptually understand the difference between rational and negative exponents ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | well , this is just going to be 5 . 5 times 5 is 25 . or you could say , 25 is equal to 5 squared . | is there such thing as a rational exponent with variable ( s ) , like 5^ ( 5r ) ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so this says hey , give me the number that if i were to multiply that number , or i were to repeatedly multiply that number five times , what is that , i would get 32 . well , 32 is the same thing as 2 times 2 times 2 times 2 times 2 . so 2 is that number , that if i were to multiply it five times , then i 'm going to ... | i get that +2 is supposed to be `` the principle root '' , but what about -2 , how does that work if the exponent is a improper negative fraction ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so the cube root of 8 , or 8 to the 1/3 , is just going to be equal to 2 . this says hey , give me the number that if i say that number , times that number , times that number , i 'm going to get 8 . well , that number is 2 because 2 to the third power is 8 . | how do i exponent the number ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | what is 64 to the 1/3 power ? well , we already know that 4 times 4 times 4 is 64 . so this is going to be 4 . | 4^-3 is also -4 x -4 x -4 , right ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | what is 64 to the 1/3 power ? well , we already know that 4 times 4 times 4 is 64 . so this is going to be 4 . | is 1/4 the same as -4 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | do a few more examples of that . what is 64 to the 1/3 power ? well , we already know that 4 times 4 times 4 is 64 . | sal explains 4 to the 1/3 power , then he explains 4 to the 1/2 , 4 to the 1/3 power is 64 & 4 to the 1/2 power is 2 , so the numerator do n't matter is it is 4 to the 1/3 power ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | what is a cubed root ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we already know 4 to the third is 64 , so this is going to be 1/64 . now let 's think about fractional exponents . so we 're going to think about what is 4 to the 1/2 power . | what if the fractional exponent is a negative ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that if i were to multiply it by itself , or if i were to have two of those numbers and i were to multiply them , times each other , that same number , i 'm going to get 4 ? well , what times itself is eq... | how do i add or subtract numbers that are powered to fractions ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . | does anyone know how to type an exponent on a macbook air ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | but either way , this is going to result in 4 times 4 is 16 , times 4 is 64 . we also know a little bit about negative exponents . so for example , if i were take 4 to the negative 3 power , we know this negative tells us to take the reciprocal 1/4 to the third . | is there such thing as a negative factional exponent ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | 5 times 5 is 25 . or you could say , 25 is equal to 5 squared . now , let 's think about what happens when you take something to the 1/3 power . | could you have multiplied the base by the exponent ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | is the principle root always positive for the square root only or all roots ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | but either way , this is going to result in 4 times 4 is 16 , times 4 is 64 . we also know a little bit about negative exponents . so for example , if i were take 4 to the negative 3 power , we know this negative tells us to take the reciprocal 1/4 to the third . | is there a major difference when it is negative ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | now , let 's think about what happens when you take something to the 1/3 power . so let 's imagine taking 8 to the 1/3 power . so the definition here is that taking something to the 1/3 power is the same thing as taking the cube root of that number . and the cube root is just saying , well what number , if i had three ... | does that mean if you have the exponent 1/3 you are taking the cube root and so on ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | do a few more examples of that . what is 64 to the 1/3 power ? well , we already know that 4 times 4 times 4 is 64 . so this is going to be 4 . | is 4^-3 = 1/4^3 = 1/64 the same idea as : 1/4/4/4 , or 1 divided by 4 divided by 4 divided by 4 = 1/64 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | so you 're starting to see something interesting . 4 to the 1/2 is equal to 2 , 2 squared is equal to 4 . so let 's get a couple more examples of this , just so you make sure you get what 's going on . | what would 4^-1/2 power be ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | but either way , this is going to result in 4 times 4 is 16 , times 4 is 64 . we also know a little bit about negative exponents . so for example , if i were take 4 to the negative 3 power , we know this negative tells us to take the reciprocal 1/4 to the third . | why is all the exponents non-negative or greater than zero ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | well , this is just going to be 5 . 5 times 5 is 25 . or you could say , 25 is equal to 5 squared . | how would you solve ( x-5 ) ^2/3 =16 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | now let 's think about fractional exponents . so we 're going to think about what is 4 to the 1/2 power . and i encourage you to pause the video and at least take a guess about what you think this is . | what do i search to find ( 1/4 ) to the x power ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | what is a cube root ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | we also know a little bit about negative exponents . so for example , if i were take 4 to the negative 3 power , we know this negative tells us to take the reciprocal 1/4 to the third . and we already know 4 to the third is 64 , so this is going to be 1/64 . | do you simply approximate/take the limit ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | but either way , this is going to result in 4 times 4 is 16 , times 4 is 64 . we also know a little bit about negative exponents . so for example , if i were take 4 to the negative 3 power , we know this negative tells us to take the reciprocal 1/4 to the third . | would the principle root of a negative have a positive or a negative one ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | these are both true statements . let 's do one more like this . what is 25 to the 1/2 going to be ? | what would happen if you made a fractional exponent with a numerator other than one , like 9^2/3 ? |
we already know a good bit about exponents . for example , we know if we took the number 4 and raised it to the third power , this is equivalent to taking three fours and multiplying them . or you can also view it as starting with a 1 , and then multiplying the 1 by 4 , or multiplying that by 4 , three times . but eith... | and we 'll talk in the future about why this is , and the reason why this is defined this way , is it has all sorts of neat and elegant properties when you start manipulating the actual exponents . but what is the square root of 4 , especially the principal root , mean ? well that means , well , what is a number that i... | how do i find the root of a number , without a calculator , if i do n't actually know what the root is ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | the n-th derivative evaluated at 0 . and that 's why it makes applying the maclaurin series formula fairly straightforward . if i wanted to approximate e to the x using a maclaurin series -- so e to the x -- and i 'll put a little approximately over here . | how would you make the maclaurin series for number pi ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | the n-th derivative evaluated at 0 . and that 's why it makes applying the maclaurin series formula fairly straightforward . if i wanted to approximate e to the x using a maclaurin series -- so e to the x -- and i 'll put a little approximately over here . and we 'll get closer and closer to the real e to the x as we k... | what is the maclaurin series for tan ( x ) ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | that if you wanted to approximate e , you just evaluate this at x is equal to 1 . so if you wanted to approximate e , you 'd say e is approximate to -- well , e is e to the first power . and that 's going to be approximately equal to this polynomial evaluated at 1 . | how would you incorporate imaginary numbers to reconcile e , cos , and sin like sal alluded to ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | it 's going to be equal to any of the derivatives evaluated at 0 . the n-th derivative evaluated at 0 . and that 's why it makes applying the maclaurin series formula fairly straightforward . | from n=0 to infinity converges ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | it 's going to be equal to any of the derivatives evaluated at 0 . the n-th derivative evaluated at 0 . and that 's why it makes applying the maclaurin series formula fairly straightforward . | why does n't sal talk about the tan taylor series at 0 ( maclaurin ) ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | and something pretty neat is starting to emerge . is that e to x , 1 -- this is just really cool -- that e to the x can be approximated by 1 plus x plus x squared over 2 factorial plus x to the third over 3 factorial . once again , e to the x is starting to look like a pretty cool thing . | how would the maclaurin series for [ ( 1+e to the x ) squared ] be found ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | so plus x plus , this is also 1 , so it 's going to be x squared over 2 factorial . so plus x squared over 2 factorial . all of these things are going to be 1 . | why does the approximation not alternate between + and - ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | and something pretty neat is starting to emerge . is that e to x , 1 -- this is just really cool -- that e to the x can be approximated by 1 plus x plus x squared over 2 factorial plus x to the third over 3 factorial . once again , e to the x is starting to look like a pretty cool thing . | can you construct the maclaurin series for f ( x ) = e ^ ( -1/x^2 ) ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , frankly , one of the amazing things about the n... | how would you find the maclaurin expansion of f ( x ) =e^3x ? |
now let 's do something pretty interesting . and this will , to some degree , be one of the easiest functions to find the maclaurin series representation of . but let 's try to approximate e to the x. f of x is equal to e to the x . and what makes this really simple is , when you take the derivative -- and this is , fr... | that if you wanted to approximate e , you just evaluate this at x is equal to 1 . so if you wanted to approximate e , you 'd say e is approximate to -- well , e is e to the first power . and that 's going to be approximately equal to this polynomial evaluated at 1 . | what is the expansion of e ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | consider the following challenge . alice and bob can transmit and receive messages binary . ( morse code ) they charge their customers 1 penny per bit to use their system , and a regular customer arrives who wants to send a message , and their messages are 1,000 symbols long . | is audio data pseudo-random from a binary point of view ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | ( morse code ) while , if the entropy increases due to unpredictability , our ability to compress decreases . ( morse code ) if we want to compress beyond entropy , we must necessarily throw away information in our messages . | is audio compression done by throwing away non-audible frequencies from a fast fourier transform ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | for example , if you shorten the code for d to just 0 , then a message 011 could perhaps mean daa , or maybe just b . so for this to work , you would need to introduce letter spaces , which cancel out any savings during transmission . now , how far does this compress the message compared to the original 2,000 bits ? | how much are letter spaces accounted for in compression codes ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | : when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . | does radio send information using binary digits ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | : when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . | or do the waves cause the receiver to vibrate at a frequency conducive for analog decompression into audio waves ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | that means with this huffman coding we can expect to compress the messages from 2,000 bits to 1,750 bits . and claude shannon was the first to claim that the limit of compression will always be the entropy of the message source . as the entropy , or uncertainty , of our source decreases due to known statistical structu... | what does `` entropy '' mean ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | david huffman famously provided the optimal strategy , which he published in 1952 , and based on building a binary tree from the bottom up . to begin , we can list all symbols at the bottom which we can call nodes . then we find the two least probable nodes , in this case b and c , and merge them into one , and add the... | how do you group the nodes when their probabilities do n't imply obvious divisions ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | normally , this is sent with a standard 2-bit code , which results in charging for 2,000 bits . however , alice and bob already did some analysis on this customer before , and determined that the probability of each symbol in the message is different . can they use these known probabilities to compress the transmission... | nevertheless are n't separators really useful for quickly accessing a certain symbol ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | normally , this is sent with a standard 2-bit code , which results in charging for 2,000 bits . however , alice and bob already did some analysis on this customer before , and determined that the probability of each symbol in the message is different . can they use these known probabilities to compress the transmission... | so are the compression algorithms different for different languages ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | however , alice and bob already did some analysis on this customer before , and determined that the probability of each symbol in the message is different . can they use these known probabilities to compress the transmission and increase their profits ? what 's the optimal coding strategy ? | is there a standard for how to assign probabilities for each language ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | : when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . | what is the prefix used to say `` this is a new character '' ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | that means with this huffman coding we can expect to compress the messages from 2,000 bits to 1,750 bits . and claude shannon was the first to claim that the limit of compression will always be the entropy of the message source . as the entropy , or uncertainty , of our source decreases due to known statistical structu... | what does it mean by the statement that limit of compression will always be the entropy of the message source ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | so we multiply the length of each code times the probability of occurrence , and add them together , which results in an average length of 1.75 bits per symbol on average . that means with this huffman coding we can expect to compress the messages from 2,000 bits to 1,750 bits . and claude shannon was the first to clai... | it looks like the symbols are only as , bs , cs , and ds ... so two bits for each ( 00 , 01 , 11 , 10 ) ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | consider the following challenge . alice and bob can transmit and receive messages binary . ( morse code ) they charge their customers 1 penny per bit to use their system , and a regular customer arrives who wants to send a message , and their messages are 1,000 symbols long . | it says that alice and bob would get more profit by compressing their code , but sending less bits means getting less pennies right ? |
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