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In $11-38,$ evaluate each expression in the set of real numbers.$$\sqrt[5]{-\frac{1}{32}}$$

$-\frac{1}{2}$

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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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In $11-38,$ evaluate each …

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All right. Hey, guys. So, in this problem, we're taking the fifth root of negative one over 32. The problem asks us to simplify this expression, and it states that are answers should be something in the real play, the real numbers. And so let's tackle this problem with them Algebraic steps, basically rewriting what we're given first. So we're given the fifth root of negative one over 32. Now, what we can do is we can do this next step where we basically split the fifth root apart into the numerator and into the denominator. So we have We're taking the fifth root of the numerator, and then we're taking the fifth root of the denominator. Now let's take a look at the numerator first, the numerator. Well, in a previous problem, well, we already did a problem like this. We said our fifth root of negative one is basically the same as fifth root of negative one to the fifth. Because in that problem, we saw the negative ones of the fifth is equal to negative one. And when you take the fifth would have something to the fifth u get net. You get that something So when you take 1/5 root of negative one to the fifth u get negative want. So from this, we can easily see that our numerator evaluates to negative one. And now we take a look at her denominator. So, like that previous problem, we want to do a similar thing we want to find. We want to cancel our fifth root out, and the way we do that is, if we take the fifth root of something to the fifth. So we want to rewrite our 32 as some number two, the fifth Power. And luckily through algebra, um, we see that 32 is actually equal to two to the fifth. So you multiply two times itself five times and you get 32. And so when you do that, fifth root of two to the fifth is going to be equal 22 and two to the fifties 32. So we know that fifth root of 32 is just equal to two. So our answer is negative. 1/2 show negative. 1/2 is the answer to the radical expression. Ah, fifth root of negative of negative one over 32. Now just to, uh I'm just gonna say this verbally because there is another way we could have tackle this problem. We could have tackle this problem through a In this problem, we tackle the problem by splitting up, splitting our radical into the numerator and denominator. But what? But what we could have also done is we could have done the fifth root of negative one times the fifth root of one over 32. And in doing this, we could have just solved each of these individually and you get negative one times one have. And then you would still hopes you would still get your answer to be, uh, negative 1/2. Just if you used this way to solve the problem. What happened is you just need to recognize that one over 32 is just equal to 1/2 to the fifth power. Just say if you get the same ends or just a different process, different approach. Thanks for listening, guys. And I hope this

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