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

(a) $ \dfrac{x^{2n} \cdot x^{3n-1}}{x^{n + 2}} $(b) $ \dfrac{\sqrt{a\sqrt {b}}}{\sqrt [3]{ab}} $

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a) $x^{4 n-3}$b) $\frac{a^{\frac{1}{6}}}{b^{\frac{1}{12}}}$

05:03

Jeffrey Payo

Calculus 1 / AB

Calculus 2 / BC

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Functions and Models

Section 4

Exponential Functions

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September 10, 2019

At the very end of your video it should be 6th root of a DIVIDED by 12th root of b.

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Okay, let's use our laws of exponents to simplify these expressions. So for the numerator, what we're going to do is add these exponents and we get X to the power of five and minus one. And then when we have a quotient, the quotient rule for exponents tells us that we need to subtract these exponents. And so that's going to give us X to the power five in minus one, minus the quantity and plus two. So let's go ahead and simplify that exponents. We have five in minus one minus and minus two when we distribute the negative sign. And so that gives us an exponents of four in minus three. So are simplified. Expression is X to the power foreign minus three now on to part B. So to simplify this one, let's go ahead and change it from radical form to exponents form. And then we can use exponents properties like we did in part a. So a square root is equivalent to a 1/2 power. So we have a square ruby to the 1/2 power, and a cube root is equivalent to a 1/3 power, so the denominator is a B to the 1/3 power. Okay, now let's also take care of the square root of be that we see inside the numerator. So we have a times B to the 1/2 power all to the 1/2 over a B to the 1/3 power. Okay, Now let's raise each of these factors to the power of 1/2 or 1/3 depending on where it is in the problem. So we have eight of the 1/2 power and then be to the 1/2 to the 1/2 we're going to multiply those exponents. That's the power property of exponents, and that gives us be to the 1/4 power. And then on the bottom, we have eight of the 1/3 power bi to the 1/3 power. Okay, so remember the quotient property that we used on part A. We're going to subtract exponents, so let's bring it over here. So we have eight of the 1/2 divided by age of the 1/3. That's going to be a to the 1/2 minus 1/3. We need to subtract those exponents and we have be to the 1/4 divided by B to the 1/3. That's going to be be to the 1/4 minus 1/3. All right, so the last thing we want to do is subtract thes fractions. 1/2 minus 1/3 that's 26 minus are 36 miles to six. That's 1/6 a to the 16 and then subtracting fractions on B. So 1/4 minus 1/3 that would be 3 12 smiles, 4/12 and that would be negative 1/12 Now, lastly, we don't typically like to leave a negative exponents. So let's change this into a to the 1/6 power over B to the 1 12 power and there we go.

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