1Cademy - Power Property for Exponents

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Exponential Notation in Algebra

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Power Property for Exponents

The Power Property for Exponents provides a shortcut for raising an exponential expression to another power. If $a$ is a real number, and $m$ and $n$ are whole numbers, then:

$(a^m)^n = a^{m \cdot n}$

In words: to raise a power to a power, keep the base and multiply the exponents. The property follows from the definition of exponents. Consider $(x^2)^3$ : the outer exponent $3$ means $x^2$ appears as a factor three times, giving $x^2 \cdot x^2 \cdot x^2$ . Expanding each $x^2$ into its two factors of $x$ reveals $2 + 2 + 2 = 6$ factors of $x$ altogether, so $(x^2)^3 = x^6$ . Because $6 = 2 \cdot 3$ , the total number of factors equals the product of the two exponents. A numerical check confirms the rule: $(3^2)^3 = 9^3 = 729$ , and $3^{2 \cdot 3} = 3^6 = 729$ ✓.

Updated 2026-05-01

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