Question

2. Simplify the given expression to rational exponent form, justify each step by identifying the properties of rational exponents used.

Answer 1

I dont want the answer, I just want to know how to start this off.

Answer 2

Start with \[
\sqrt[n]{x} = x^{1/n}
\]

Answer 3

So \[
\sqrt[3]{x^{-6}} = \left(x^{-6}\right)^{1/3}
\]

Answer 4

Does that make sense?

Answer 5

Well, how did you get rid of the radical?

Answer 6

This is the pattern \[
\sqrt[\color{blue}{n}]{\color{red}{x}} = \color{red}{x}^{1/\color{blue}{n}}
\]

Answer 7

So \[
\sqrt[\color{blue}{3}]{\color{red}{x^{-6}}} = \left(\color{red}{x^{-6}}\right)^{1/\color{blue}{3}}
\]

Answer 8

Can you understand the pattern?

Answer 9

I think so. Thanks.

Answer 10

What about that numerator 1 over the radical equation part.

Answer 11

Well \[
\frac{1}{\color{red}x} = \color{red}x^{-1}
\]

Answer 12

But first we need to simplify \[
({x^a})^b = x^{a\times b}
\]

Answer 13

So \[
\left(x^{-6}\right)^{1/3} = x^{-6\times 1/3}
\]

Answer 14

And \(-6\times 1/3 = -2\) so \[
x^{-6\times 1/3} = x^{-2}
\]

Answer 15

We have simplified it a bit: \[
\frac{1}{\sqrt[3]{x^{-6}}} =\frac{1}{x^{-2}}
\]

Answer 16

Does this part make sense?

Answer 17

I see. Yes, it does. I just need to learn the pattern.

Answer 18

Can you do the rest?

Answer 19

Yes. Thank you ^.^

Answer 20

The answer would be x^2 right?

Answer 21

Yes

Answer 22

Thank you for your help.