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. ∛2/3

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression, we're taking the third rut of a fraction in the numerator we have four. And in the denominator we have seven. We need to rationalize this, we need to simplify it such that there is no rad in the denominator. New radicals in the denominator allowed. Will we recall from previous lessons that due to the quotient rule of radicals, we can rewrite this as taking the third root of both. Our numerator. So the third rate of four and the third root of our denominator. So the third root of seven so that we can concentrate on getting that seven out of that third rut. Well how do we do that? Well, in order for anything to come out of a radical with an index of three, there has to be three of that factor, there's only 17. So we need two more. We need to multiply this by the third rut of another seven times seven or seven times seven is 49. So we're going to multiply both the numerator and the denominator by the third rut of 49 or seven times seven. So that we are in effect multiplying it by one. Well now let's just do that multiplication in the numerator. We have the third rut of four times seven times seven, there aren't three of any single factor up there so nothing will be able to come out. But in our denominator we're taking the third root of now seven times seven times seven there are three sevens. So it's like we've got our lucky sevens and gets to come out. So our denominator is now just seven In the numerator. We can multiply the four times the seven times the seven. So we're taking the third root of 196. That is as simplified as it gets. We can't take anything else out in the numerator because there aren't three of any factors. So this is our final answer. We have the third rate of 196 divided by seven. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

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Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∛2/3

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