In Exercises 29–44, simplify using the quotient rule._______√x²/1...

Question

In Exercises 29–44, simplify using the quotient rule._______√x²/144y¹²

Accepted Answer

Welcome back. I am so glad you're here. We are asked to use the quotient rule for the simplification of the square root of. We have a fraction. In the numerator, we have nine H squared and in the deno that's divided by 361 K rays to the 20th power. Our answer choices are answer choice. A three H rays to the 10th power divided by 19 K. Answer choice B 19 H divided by three K race to the 10th power. Answer choice C three H divided by 19 K raise to the 20th power and answer choice D three H divided by 19 K rays to the 10th power. All right, the quotient rule tells us that whenever we're taking the rut of a fraction, we can rewrite that, taking the rut of both the numerator and the denominator. So in this case, we're taking the square root of our numerator nine H squared Divide that by the square root of our denominator. 361 K raises to the 20th power. And now we can focus on simplifying both our numerator and our denominator. We'll start with the numerator. We first look when simplifying radicals to see what the index of our radical is. And when there's nothing written, it's going to be an index of two. So that means that we need to have two of any factor in order for it to come outside of our radical. Let's look first at the nine, does the nine have two of any factor? It does, it has two, threes nine is three multiplied by three or in other words, three squared. So we can rewrite this numerator instead of having a nine inside of our square root, we can put three squared. And now taking a look at our H squared. Well, that one is already written as something with a square that is H squared. And now let's take a look at our denominator. We have 361, 361 is 19 multiplied by 19 or in other words, 19 squared. So we can put that underneath the radical. We have the square root of no longer 361. But now 19 squared. And now taking a look at this K raised to the 20th power. We recall from previous lessons that we could rewrite that as K raised to the 10th power multiplied by K raised to the 10th power because when we are multiplying like bases, those are the K S. In this case, we can add the exponents and 10 plus 10 equals 20. And we know that K raised to the 10th power multiplied by K raised to the 10th power is the same as K raised to the 10th power Squared. So back underneath our radical in the denominator, we have the square root of 19 squared multiplied by no longer K raised to the 20th power, but now K raised to the 10th power squared. And so now anything that is squared gets to come outside of the radical. So in the numerator, that'll be the three and the h and when they come outside of the radical, they lose, they're squared. So that will just be three h in our numerator. And that's divided by the things that are squared. And the denominator get to come out of the radical that's 19 and K raised to the 10th power. So we'll have 19 K raised to the 10th power in the denominator. And this is as simplified as this one gets, we look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

In Exercises 29–44, simplify using the quotient rule._______√x²/144y¹²

Key Concepts

Quotient Rule

Radical Expressions

Simplification of Fractions

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