41.4 Integer Exponent Product and Quotient Rules

Integer Exponent Product and Quotient Rules describes how powers sharing the same base combine under multiplication and division, by adding exponents for a product and subtracting exponents for a quotient, extending naturally to zero and negative integer results.

Integer Exponent Product Rule

Rule Statement

When multiplying two powers that share the same base, the exponents are added together while the base remains unchanged:

Example

This rule applies regardless of whether $m$ and $n$ are positive, zero, or negative integers.

Integer Exponent Quotient Rule

Rule Statement

When dividing two powers that share the same base, the exponent of the denominator is subtracted from the exponent of the numerator, while the base remains unchanged:

Example

Exponent Difference with a Positive Result

Description

When applying the quotient rule, if the numerator's exponent is greater than the denominator's exponent, the subtraction produces a positive result, and the simplified expression is written directly as a positive power.

Example

Exponent Difference Equal to Zero

Description

When applying the quotient rule, if the numerator's exponent equals the denominator's exponent, the subtraction produces a result of zero, and the zero-exponent rule applies, giving a final simplified value of one for any nonzero base.

Example

Exponent Difference with a Negative Result

Description

When applying the quotient rule, if the numerator's exponent is smaller than the denominator's exponent, the subtraction produces a negative result, and the negative-exponent rule applies, converting the expression into a reciprocal with a positive exponent.

Example

Common Base Requirement

Requirement

Both the product rule and the quotient rule require that the two powers involved share exactly the same base; the rules do not apply directly to powers with different bases.

Example of Nonapplicability

An expression such as $x^3·y^2$ cannot be combined into a single power using these rules, since $x$ and $y$ are different bases.

Quotient Denominator Restriction

Requirement

In the quotient rule, the base in the denominator must be nonzero, since a zero base in the denominator would make the entire expression undefined regardless of the exponent rules being applied.

Reasoning

This restriction is consistent with the broader requirement that division by zero is never a valid operation, regardless of what exponent is attached to the zero base.