1Cademy - Simplifying $$(m^4n^{-3})(m^{-5}n^{-2})$$ and $$(p^6q^{-2})(p^{-9}q^{-1})$$ Using the Product Property

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Negative Exponent

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Simplifying $(m^4n^{-3})(m^{-5}n^{-2})$ and $(p^6q^{-2})(p^{-9}q^{-1})$ Using the Product Property

Apply the Commutative Property and the Product Property for Exponents to simplify products containing multiple variables with negative exponents.

ⓐ $(m^4n^{-3})(m^{-5}n^{-2})$ : Use the Commutative Property to group factors with the same base together: $m^4m^{-5} \cdot n^{-3}n^{-2}$ . Apply the Product Property by adding the exponents for each base: for $m$ , $4 + (-5) = -1$ ; for $n$ , $-3 + (-2) = -5$ . The expression simplifies to $m^{-1}n^{-5}$ . Finally, apply the definition of a negative exponent to both variables, moving them to the denominator: $\frac{1}{m^1} \cdot \frac{1}{n^5} = \frac{1}{mn^5}$ .

ⓑ $(p^6q^{-2})(p^{-9}q^{-1})$ : Group like bases: $p^6p^{-9} \cdot q^{-2}q^{-1}$ . Add the exponents for $p$ ( $6 + (-9) = -3$ ) and for $q$ ( $-2 + (-1) = -3$ ), resulting in $p^{-3}q^{-3}$ . Take the reciprocals to change the signs of the exponents, yielding $\frac{1}{p^3q^3}$ .

When dealing with multiple variables, group identical bases first, sum their exponents independently, and then resolve any resulting negative exponents by moving those specific factors to the denominator.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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