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. ∛8/x⁴

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression, we have the third row of this whole fraction. In the numerator it's 27 in the denominator it's X to the seventh. We are asked to simplify this expression noting that all the variables represent positive real numbers. So we don't have any possible domain issues. So let's recap call from previous lessons that when we are taking the third rut of a fraction, we can rewrite this because of the quotient rule of radicals. We can rewrite this as taking the third rut of our numerator, third of 27 divided by the third rut of our denominator. The third root of X to the seventh. And now we can simplify where we can. Well the numerator, the third rut of 27 is going to be three. And the denominator the third root of X to the seventh. We need to get those X to the seventh out of there because we can't have any roots in our denominator. So we can break this X to the seventh apart. There's an X to the third times and X to the third times an X to the first. Because if you add all of these together, that would equal the seventh and the first two can come out. So that's an X squared. That gets to escape. That is very exciting. And there's still an X to the first that is having its cube taken and it cannot come out. It is very sad, it is stuck in there. So we need to help it. How do we help it? Well it needs two more exes in order to get out of there so we can multiply both the numerator and the denominator by X squared. And it's going to be the cubed writ of X squared in order to get that X to the first out of there. So simplifying this in the numerator we've got three times the cubed rut of X squared. That X squared is trapped. Sorry. And in the denominator we've got an X squared out in front and we have the cubed rut of X to the first times X squared which is X to the third which will get to come out. So now we have an additional X coming out. Our denominator is now X to the third. The numerator remains three times the cubed root of X squared. Well this is as simplified as it gets. So we look at our answer choices for three times the cubed of X squared all over X cubed. And that matches with answer choice. A Well 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. ∛8/x⁴

Key Concepts

Radical Expressions

Properties of Exponents

Simplification of Fractions

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