How to raise a fraction to a fractional exponent?

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I am trying to simplify this but I'm not sure how to approach it.

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5 Answers

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Write $\frac{1}{9} = 3^{-2}$.

Then

$\big(\frac{1}{9}\big)^{\frac{3}{2}} = 3^{-2 \cdot \frac{3}{2}} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27}$.

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Try applying logarithms:

$a^b = c \implies b \ln a = \ln c$

and you also know that $1^x=1$.

Good luck!

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Hint: this is equivalent to $\sqrt{\left(\frac{1}{9}\right)^3}=\sqrt{\frac{1}{729}}$

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$(\frac{1}{9})^{\frac{3}{2}}=(\sqrt{\frac{1}{9}})^3=\frac{1}{27}$

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In general, if $a>0$, the following will hold:

$$a^{\frac pq}=(\sqrt[q]{a})^p.$$

In your case:

$$(\frac19)^{\frac32}=(\sqrt[2]{\frac19})^3=(\frac13)^3=\frac{1}{27}.$$

What about $a<0$? Well, if the power is defined, the above will hold.

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