1Cademy - Simplifying $$a^7 \cdot a$$ and $$x^{27} \cdot x^{13}$$ Using the Product Property

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Product Property for Exponents

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Simplifying $a^7 \cdot a$ and $x^{27} \cdot x^{13}$ Using the Product Property

Apply the Product Property for Exponents to two expressions with variable bases.

ⓐ $a^7 \cdot a = a^8$ : The factor $a$ has no visible exponent, so rewrite it as $a^1$ — any base written without an exponent is understood to carry an exponent of $1$ . Now both factors share the base $a$ , and the product property applies: $a^7 \cdot a^1 = a^{7+1} = a^8$ .

ⓑ $x^{27} \cdot x^{13} = x^{40}$ : Both factors already have the same base $x$ , so add the exponents directly: $x^{27} \cdot x^{13} = x^{27+13} = x^{40}$ .

Part (a) reinforces a key detail: a variable written without an exponent implicitly has an exponent of $1$ , which allows the Product Property to be applied directly. Part (b) shows the property works the same way regardless of how large the exponents are — simply add them.

Updated 2026-04-21

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OpenStax Elementary Algebra

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