Perform the indicated operations and/or simplify each expression....

Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∜(g³h⁵/9r⁶)

Accepted Answer

Welcome back. I am so glad you're here we are Given this big expression we're taking the fourth root of in the numerator. It's A to the fifth B to the sixth and that's all over 25 C to the 10th. We need to simplify this expression noting that the variables all represent positive real numbers so we don't have any potential domain issues. So how do we simplify this? Will we recall from previous lessons that when we're taking the rut of a fraction we can break it down and take the root of the numerator and the denominator. So we can take the four threat of the numerator A to the fifth, B to the sixth, divided by the four threat of the denominator. 25 C to the 10th. And now I'm going to start by focusing on the denominator. We need to get that route out of there. We can't have roots in the denominator so we need to get the 25 seat of the 10th out. There needs to be four of whatever factor in order for it to come out. Well 25 is already five times five so that's halfway there. They need another five times five to get out and that seat of the 10th, we could break that down that seat of the fourth times C to the fourth that gets us to eight times another C squared. That gets us to our 10. But that C squared won't be able to come out without another two CS. So we multiply times another C squared. So we're going to be multiplying both the numerator and the denominator by the fourth root of five times five, which is 25. And then R. C squared. So multiplying both the numerator and the denominator by the fourth threat of 25 C squared. All right now looking at this denominator, writing it all out, we've got the fourth root of we've got five times five times five times 51234 of them. Yeah, A five can come out. So that's in our new denominator and we've got seed of the fourth seed of the fourth and one last seat of the fourth which came from R. C squared times C squared. So three seat of the fourths come out, that's C to the third. There's three CS coming out and in our numerator now we have got the fourth threat of all of this. We've got I'll write these in order starting with the coefficients. So 25 A to the fifth, B to the sixth, C squared. Now let's break down that numerator a bit. We'll start on getting what we can out of that four threat. So writing this out where we can with things that come in fours 25 that's just five times five. We're not going to get enough five to come out. Eight of the fifth. That's the same as A to the fourth times eight of the first B to the sixth. That's the same as B to the fourth times b squared C squared is just C squared. It's not going to have enough to come out. But look, we can take out this A to the fourth and this B to the fourth, so that will be an A. And a B coming out in front. And we've still got the four threat of everything else left in there. So that's 25 8 of the first. We can just write a B squared C squared all over five seat of the third. This A B and this five seat of the third. They don't have any factors in common. So this is as simplified as it's going to get. So we're looking at our answer choices for A. B. Four threat of 25 A B squared C squared in the numerator, all divided by five C to the third that matches with answer choice C. While done, we'll catch you on the next one.

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∜(g³h⁵/9r⁶)

Key Concepts

Radical Expressions

Exponents and Powers

Simplifying Fractions

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