Use the product rule to simplify the expressions in Exercises 13–...

Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √10x⋅√ 8x

Accepted Answer

Welcome back. I'm so glad you're here. We have got an expression and we're going to simplify it using the product rule. Well what is the product rule? The product rule for radicals is that when we have the square root of two terms A times B. That equals the square root of A. Times the square root of B. And likewise, when we have the square root of a number A times the square root of a different number B. That is the same as the square root of A. Times B. All under the same radical. So what have we got here? We've got the square root of 25 A. Times The Square Root of four. A. And which example does that look like from our rule? It looks like this second one here, the 25 A. Is like our A. And the four A. Is like R. B. And now let's combine those under one radical. So we've now got the square root of 25 A. Times four A. And I'm going to rewrite that with our whole numbers together. So we've got 25 times four times a times a. And now let's do the multiplication under there. We've got the square root of 25 times four is 108 times a. Is a square. Well we know that we could rewrite 100 as 10 squared. So we've got the squared of 10 squared times a squared. And this is great because the unwritten index here is an understood too. That too is like a gatekeeper and he says there have to be two of whatever factor in order for you to escape from this radical. And the 10 and the I get really excited because they're like there's two of us, there's two of us to tens and two aces. And the gatekeeper says you're right, you can come out. So so they go through the gate and when they go through that gate now there are no longer two tens, but there is 1 10 and no longer to A. S. But there is one A. And so our final answer here is 10 A. We look at our answer choices and that matches with answer choice. D. Well done. We'll catch you on the next one.

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