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Use the Law of Exponents to rewrite and simplify the expression.

(a) $ \dfrac{4^{-3}}{2^{-8}} $

(b) $ \dfrac{1}{ \sqrt[3]{x^4}} $

a) 4

b) $\frac{1}{x \sqrt[3]{x}}$

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Missouri State University

Harvey Mudd College

Baylor University

Idaho State University

All right, let's use laws of exponents to simplify these expressions. So first of all, we notice that for part A, you could write four as a power of two. So four can be written as two squared. So we have to square to the negative third power over to to the negative eighth power. And we can simplify this because we haven't exponents rule that says when you have power to a power, you multiply them. This will be to to the negative sixth power over to to the negative eighth power. Now, the next exponents rule we want to recall is that we have a quotient rule that says you should subtract the exponents. So we have to to the power of negative six minus negative eight. Subtract thes exponents here. Negative six minus negative. Eight is too. So we have two squared and that is just four for Part B. Noticed that inside the radical you have X to the fourth, but it's a cube root. So what if we break it down into the cube? Root of X Cubed times X. Since X cubed times X is X to the fourth. Now the cube root of X cubed is just going to be X so we can simplify the radical by putting the X outside the route. And we still have the single X inside the route. So we get one over X cube root X.