In Exercises 45–66, divide and, if possible, simplify.
______
⁵√9...

Question

In Exercises 45–66, divide and, if possible, simplify.
______
⁵√96x¹²y¹¹
⁵√3x²y⁻²

Accepted Answer

Welcome back. I am so glad you're here. We're asked for the following expression, perform division and further simplify if possible. The expression we're given is in the numerator, we're taking the fifth threat of M raise to the 17th K squared. That's divided by, in the denominator, we're taking the fifth rut of two M raised to the 12th power. K raised to the negative fourth power. Our answer choices are answer choice. A three M fif R M K answer trace B three K fifth rut M K, answer choice C three M K fifth threat K and answer choice D three M K fifth threat M All right. So the first thing we notice here is that we're taking the fifth rut of both our numerator and our denominator. And we recall from previous lessons that when we're taking that same rut of both the numerator and the denominator, we can rewrite it, taking the rut of the whole fraction. So in this case, we're taking the fifth rut of in our numerator, it's 486 M raised to the 17th K squared. And in our denominator, we have two M raised to the 12th K, raise to the negative fourth. And now we just need to, to divide the parts that are alike. We can start with our coefficients. 486 divided by 2, 486 divided by two is 243. So we can rewrite this as now the fifth threat of starting with 243. Now we can look at our MS. We have M race to the 17th divided by M rays to the 12th power. We recall from previous lessons that when we are dividing like bases, those are the MS, we are going to subtract their exponents. So we take 17 minus and 17 minus 12 equals five. So that's our new exponents here. This M raise to the 17th divided by M raise to the 12th is equal to M raise to the fifth. We can put that inside of our fifth rut. Now for the final part here, we have K squared divided by K raised to the negative fourth. Well, we have like bases here so we can subtract our exponents and we're taking two minus negative four, two minus negative four is the same as two plus four. So that equals six. So when we divide these KS, that gives us K raise to the sixth power and we can put that under the radical. So we're taking the fifth threat of 243 M raised to the fifth K raised to the sixth and now we look at our index, it's five that tells us that we need five of any factor in order for it to come outside of the radical. Well, 243 that's the same as three raised to the fifth power And M raised to the 5th power that's also raised to the 5th power so far. This is looking pretty good. However, that K race to the sixth power, well, that's to the sixth, not to the fifth. We recall from previous lessons that we could rewrite K race to the sixth power as K raised to the fifth power multiplied by K or which is K raise to the first power. Because when we have like bases, the KS and they're being multiplied, then we are adding the exponents and five plus one equals six. So if we rewrite this under the radical as K raised to the fifth multiplied by K, then we have something else that's raised to the fifth power. Now, anything that's raised to the fifth power can come outside of the radical and when it does, so it loses that fifth power. So coming out of the radical will be the three, the M and one of the KS. So we'll have three M K and then this other K here, that one is just to the power of one, it can't come outside of the radical. So we're still taking the fifth rut of that final K and this is our answer as simplified as it gets, we look at our answer choices and this matches answer choice. C well done. We'll catch you on the next one.

In Exercises 45–66, divide and, if possible, simplify. ______ ⁵√96x¹²y¹¹ ⁵√3x²y⁻²

Key Concepts

Radical Expressions

Exponents and Their Properties

Simplifying Fractions

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