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Apply the power-of-a-quotient rule

The power-of-a-quotient rule expresses that, for a nonzero denominator and an integer exponent, (ab)n=anbn\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}: the exponent applies to both numerator and denominator, not only to the denominator or to the fraction as an unstructured symbol. This supports simplifying numerical and algebraic expressions, including scientific notation, while preserving sign and zero-exponent conventions; more advanced definitions involving noninteger or complex exponents are outside this scope.

Detailed Explanation: Apply the power-of-a-quotient rule

For a fraction with a nonzero denominator, raise both the numerator and denominator to the exponent:

(ab)n=anbn\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}

The exponent does not apply only to the denominator or to just one part of the fraction.

Example

Simplify:

(−35)2\left(\frac{-3}{5}\right)^2
  1. Apply the exponent to the numerator and denominator:
(−35)2=(−3)252\left(\frac{-3}{5}\right)^2=\frac{(-3)^2}{5^2}
  1. Evaluate each power:
(−3)2=9and52=25(-3)^2=9 \qquad\text{and}\qquad 5^2=25
  1. Write the simplified fraction:
(−35)2=925\left(\frac{-3}{5}\right)^2=\frac{9}{25}

The negative sign disappears because the numerator is raised to an even power. Remember: the exponent must be applied to both parts of the quotient.

Learn by doing: Apply the power-of-a-quotient rule

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