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Apply the power-of-a-quotient rule

For a nonzero denominator and a positive whole-number exponent, the power of a quotient satisfies (ab)n=anbn\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}: the exponent applies to both numerator and denominator because the quotient represents repeated multiplication of the entire fraction. This supports equivalent rewriting and simplification of numerical fractions and algebraic expressions, while distinguishing the rule from applying the exponent to only one part; negative or fractional exponents and more advanced domain considerations are not included.

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

When a fraction is raised to a positive whole-number power, the exponent applies to both the numerator and the denominator:

(ab)n=anbn,b≠0\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}, \qquad b\ne 0

This works because the entire fraction is multiplied by itself nn times.

Example:

(34)2\left(\frac{3}{4}\right)^2

Step 1: Apply the exponent to the numerator and denominator.

(34)2=3242\left(\frac{3}{4}\right)^2=\frac{3^2}{4^2}

Step 2: Evaluate each power.

3242=916\frac{3^2}{4^2}=\frac{9}{16}

Therefore,

(34)2=916\boxed{\left(\frac{3}{4}\right)^2=\frac{9}{16}}

The exponent applies to the whole fraction, not just one part. So (34)2\left(\frac{3}{4}\right)^2 is not 324\frac{3^2}{4} or 342\frac{3}{4^2}.

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

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