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Simplifying \left(\frac{3}{5} ight)^{-3} and \left(-\frac{a}{b} ight)^{-4} Using the Quotient to a Negative Exponent Property
Apply the Quotient to a Negative Exponent Property to simplify expressions with negative exponents applied to fractions.
ⓐ \left(\frac{3}{5} ight)^{-3} = \frac{125}{27}: The base is the fraction and the exponent is . Use the property \left(\frac{a}{b} ight)^{-n} = \left(\frac{b}{a} ight)^n to take the reciprocal of the base and change the exponent to positive, resulting in \left(\frac{5}{3} ight)^3. Evaluate the power for both the numerator and denominator: .
ⓑ \left(-\frac{a}{b} ight)^{-4} = \frac{b^4}{a^4}: The base is and the exponent is . Taking the reciprocal and switching the exponent to positive gives \left(-\frac{b}{a} ight)^4. When raising a negative base to an even power, , the result is positive: .
These examples reinforce the procedure of first inverting the fraction to handle the negative exponent, and then applying the positive exponent to both the numerator and denominator, carefully managing negative signs based on whether the exponent is even or odd.
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Ch.5 Polynomials and Polynomial Functions - Intermediate Algebra @ OpenStax
Algebra
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Simplifying \left(\frac{3}{5} ight)^{-3} and \left(-\frac{a}{b} ight)^{-4} Using the Quotient to a Negative Exponent Property