1Cademy - Simplifying $$\left(\frac{2}{3}<br>ight)^{-4}$$ and $$\left(-\frac{m}{n}<br>ight)^{-2}$$ Using the Quotient to a Negative Exponent Property

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Quotient to a Negative Exponent Property

Example

Simplifying $\left(\frac{2}{3} ight)^{-4}$ and $\left(-\frac{m}{n} ight)^{-2}$ Using the Quotient to a Negative Exponent Property

Apply the Quotient to a Negative Exponent Property to simplify two expressions — one with a numerical fraction and one with an algebraic fraction containing a negative sign.

ⓐ $\left(\frac{2}{3} ight)^{-4} = \frac{81}{16}$ : The base is the fraction $\frac{2}{3}$ and the exponent is $-4$ . Apply the property $\left(\frac{a}{b} ight)^{-n} = \left(\frac{b}{a} ight)^n$ : take the reciprocal of the fraction and make the exponent positive, yielding $\left(\frac{3}{2} ight)^4$ . Evaluate by raising both the numerator and denominator to the fourth power: $\frac{3^4}{2^4} = \frac{81}{16}$ .

ⓑ $\left(-\frac{m}{n} ight)^{-2} = \frac{n^2}{m^2}$ : The base is $-\frac{m}{n}$ and the exponent is $-2$ . Apply the property: take the reciprocal and change the sign of the exponent, which gives $\left(-\frac{n}{m} ight)^2$ . Raise each component to the second power. Because the exponent $2$ is even, the negative sign is eliminated: $(-1)^2 \cdot \frac{n^2}{m^2} = \frac{n^2}{m^2}$ .

Part (a) demonstrates the process with a positive numerical fraction, while part (b) shows that when a negative variable fraction is raised to an even negative power, the final result becomes positive after taking the reciprocal and squaring.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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