Simplify the radical expressions in Exercises 67–74, if possible....

Question

Simplify the radical expressions in Exercises 67–74, if possible. ⁵√64x^6/⁵√2x

Accepted Answer

Hello. Today we're going to be simplifying a radical expression. So the radical expression given to us is the fourth root of 81 times a to the fifth power over The 4th Root of 256 Times A. So before we continue, let's go ahead and rewrite both of these numbers as a product of prime numbers. So 81 could be rewritten as nine times nine And both nines could be rewritten as three times 3. Three are prime numbers Olga and circle those. And putting this back into the fourth root is going to give me three times itself four times or three to the fourth power furthermore, A to the fifth could be rewritten as a to the fourth power times A. So I'll go ahead and rewrite that into the fourth root. Now let's go ahead and reduce 256 into a product of prime numbers. 256 could be rewritten as four times 4 could be rewritten as two times two. So I'll go ahead and circle that for now, 64 could be rewritten as eight times And both AIDS could be written as four times 2. two is a prime number. So I'll go ahead and circle that. And both 4s could be rewritten as two times 2. And those are prime numbers. Saw circle those for now. Now putting this back into the 4th root, we get the fourth root of two times itself 12345678 times. So we get to to the eighth power. But this could also be written as two to the fourth, times two to the fourth. So I'll go ahead and write that as to the fourth times two to the fourth and that will be multiplied by eight. So our radical expression simplifies To the 4th root of three to the fourth times A. To the fourth times A. Over the fourth root Of 2 to the 4th times 2 to the 4th times a. Now the fourth root allows us to take out any number multiplied by itself four times as a single number in this case. Here we can take out three to the fourth as a single three A to the fourth at the single A. And both 2s as two. So this gives us three times a times the fourth root of a over two times two times the fourth root of a. Now notice that we have the fourth root of a. In the numerator and the denominator. So we can go ahead and cancel out that quantity that leaves us with three A over two times two and two times two is four. So that gives us three A over four. And since that's the simplest form of this of this expression, this is going to be our solution which also means that the solution to this problem is going to be a. So I hope this video helps you learn how to simplify radical expressions and I'll go ahead and see you all in the next video.

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