Question

Simplify the given expression to rational exponent form, justify each step by identifying the properties of rational exponents used. All work must be shown.

Answer 1

\[\frac{ 1 }{ \sqrt[3]{x}^-6 }\]

Answer 2

thats a 3 and a -6

Answer 3

@DebbieG

Answer 4

@jim_thompson5910

Answer 5

@wio

Answer 6

So it's:
\[\Large \dfrac{ 1 }{ \sqrt[3]{x}^{-6} }\] right?

Answer 7

I would start by changing the root expression in the den'r to a rational number. Remember: the root is the den'r, the power is the num'r:

\(\Large \sqrt[3]{x}=x^{1/3}\)

and:

\(\Large \sqrt[3]{x}^n=(x^{1/3})^n=x^{n/3}\)

Answer 8

Use all that in the den'r, and then use the fact that:

\(\Large \dfrac{1}{x^{-a}}=x^a\)

Answer 9

-6 is inside the square root tho

Answer 10

ohhhh so if \[\frac{ 1 }{ x^{-a} } = x{a} then the answer would be...\]

Answer 11

then the answer would be.. hold on

Answer 12

\[\sqrt[3]{x}^{6}\]

Answer 13

It doesn't really matter if the power is inside the root or outside. :) Why? Because:

\(\Large (x^{1/3})^n=(x^n)^{1/3}=x^{n\cdot (1/3)}=x^{n/3}\)

Answer 14

ohhh i see i see

Answer 15

You are correct in that you can take the den'r to the top (so it isn't a fraction anymore).

However, what you have there is NOT rational exponent notation, so you aren't finished. You still have the root in radical notation. Change it to a rational exponent, and simplify. :)

Answer 16

\(\Large \dfrac{ 1 }{ \sqrt[3]{x^{-6}} }=\sqrt[3]{x^{6}}\)

Now put THAT into rational exponent form.
\(\Large (x^{6})^?\)