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In $11-38,$ evaluate each expression in the set of real numbers.$$\sqrt[3]{\frac{1,000}{a^{3}},} a \neq 0$$

$\frac{10}{a}$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 2

Roots and Radicals

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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Hey, guys. So in this problem, we're given the cube root of 1000 over a cube, and we're also given the condition that are a is not equal to zero, which is important because our A's in the denominator and this problem are variable A. So if it was equal to zero, this value would be undefined. So our parameter are given condition right here is just to make sure that our function is defined, that's all. So we're being asked to compute the real number real value expression for Cuba of 1000 over a tube. So let's start that. So we have the original expression. Just gonna rewrite that over here. And so what I'm gonna do is I'm gonna split the cube root into the numerator and the denominator. So you have the cube root of 1000 in the numerator and the cube root of a cute and the denominator. Now, when when you're dealing with roots, we want to take on cube roots. We want to get rid of the cube root, and the way we do that is if we take the cube root off something cute and unfortunately for us, our denominator is already written in a form of something Cubans were in as a cute, so cute brute of a cube is just a And now we want to write our numerator, which is 1000 as something cute. And luckily for us, 1000 is also a perfect cube. It's equal to 10 cubed 10 times 10 is 100 100 times 10 is it 1000. So uh 1000 is equal to 10 cube Soren numerator is equal to uhm cube, root of 10 cute and key route of 10 Cube is the same thing is 10 So our numerator is equal to 10 and this is our answer right here. Our cube root of 1000 over a cute is just equal to 10 over a under the stipulation that are a is never equal to zero Because, like I said, if the AI was equal to zero, this value would be undefined. Thank you for listening, guys. And I hope this helps solve this problem

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