1Cademy - Simplifying $$x^{-4} \cdot x^6$$, $$y^{-6} \cdot y^4$$, and $$z^{-5} \cdot z^{-3}$$ Using the Product Property

Learn Before

Product Property for Exponents

Example

Simplifying $x^{-4} \cdot x^6$ , $y^{-6} \cdot y^4$ , and $z^{-5} \cdot z^{-3}$ Using the Product Property

Apply the Product Property for Exponents to three expressions that multiply powers of the same base when one or both exponents are negative. When the resulting exponent is negative, use the negative exponent definition to rewrite the answer with only positive exponents.

ⓐ $x^{-4} \cdot x^6 = x^2$ : Both factors share the base $x$ . Add the exponents using the Product Property: $x^{-4} \cdot x^6 = x^{-4+6} = x^2$ .

ⓑ $y^{-6} \cdot y^4 = \frac{1}{y^2}$ : Both factors share the base $y$ . Add the exponents: $y^{-6} \cdot y^4 = y^{-6+4} = y^{-2}$ . Since a negative exponent indicates the expression is not in simplest form, apply the definition $a^{-n} = \frac{1}{a^n}$ to rewrite $y^{-2}$ as $\frac{1}{y^2}$ .

ⓒ $z^{-5} \cdot z^{-3} = \frac{1}{z^8}$ : Both factors share the base $z$ , and both exponents are negative. Add the exponents: $z^{-5} \cdot z^{-3} = z^{-5+(-3)} = z^{-8}$ . Apply the negative exponent definition to obtain $\frac{1}{z^8}$ .

Part (a) shows that adding a negative and a positive exponent can yield a positive result, requiring no further simplification. Parts (b) and (c) demonstrate that when the sum of the exponents is negative, the negative exponent definition is needed to express the result with only positive exponents.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After