How do you simplify a fraction raised to a negative exponent?

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$\left( \frac23 \right) ^{-2}$ I noticed that the answer is $\frac94$, and I can't come to the conclusion of why it is?

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3 Answers

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Guide:

Approach $1$:

Well, first evaluate $\left( \frac23\right)^{-1}= \frac{a}{b}$

After which you can compute $\left(\frac{a}{b}\right)^2=\frac{a^2}{b^2}$

Approach $2$:

Well, first evaluate $\left( \frac23\right)^{2}= \frac{c}{d}$

After which you can compute $\left(\frac{c}{d}\right)^{-1}=\frac{d}{c}$

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$$x^{-1}=\frac 1x$$

Thus $$\left(\frac 23 \right)^{-2}=\left(\left(\frac 23\right)^2\right)^{-1}=\left(\frac 49 \right)^{-1}=\frac 94$$

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Because $\;\displaystyle\Bigl(\frac{a}{b}\Bigr)^{-2}=\biggl(\Bigl(\frac{a}{b}\Bigr)^{-1}\biggr)^2=\Bigl(\frac{b}{a}\Bigr)^2=\frac{b^2}{a^2}.$

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