Skill: Apply exponent laws for quotients

Explanation and Free Practice Resources

For a nonzero common base, the quotient of powers is found by subtracting exponents: $\frac{a^m}{a^n}=a^{m-n}$ , because common factors in the numerator and denominator cancel. This applies to numerical and simple algebraic expressions with whole-number exponents, including zero exponents and, when the result is negative, reciprocal form such as $a^{-k}=\frac{1}{a^k}$ ; it does not extend here to zero bases or more advanced exponent generalizations.

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Detailed Explanation: Apply exponent laws for quotients

When dividing powers with the same nonzero base, keep the base and subtract the exponent in the denominator from the exponent in the numerator:

\frac{a^m}{a^n}=a^{m-n}

Example:

\frac{5^3}{5^5}

Both powers have base $5$ , so subtract the exponents:

\frac{5^3}{5^5}=5^{3-5}=5^{-2}

A negative exponent means take the reciprocal:

5^{-2}=\frac{1}{5^2}=\frac{1}{25}

Therefore,

\boxed{\frac{5^3}{5^5}=\frac{1}{25}}

This works because the common factors cancel:

\frac{5\cdot5\cdot5}{5\cdot5\cdot5\cdot5\cdot5} =\frac{1}{5\cdot5} =\frac{1}{25}

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Exponents - Division - Positive by Positive to Positive

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