1Cademy - Deriving $$8^{\frac{1}{3}} = \sqrt[3]{8}$$ Using the Power Property

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Power Property for Exponents

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Deriving $8^{\frac{1}{3}} = \sqrt[3]{8}$ Using the Power Property

Use the Power Property for Exponents to find a value $p$ such that $(8^p)^3 = 8$ , thereby showing that $8^{\frac{1}{3}} = \sqrt[3]{8}$ .

Start with $(8^p)^3 = 8$ .

Apply the Power Property. Multiply the exponents on the left side: $(8^p)^3 = 8^{3p}$ . The equation becomes $8^{3p} = 8$ .
Write the exponent on the right. Since $8 = 8^1$ , the equation is $8^{3p} = 8^1$ .
Set the exponents equal. Because the bases are the same, the exponents must match: $3p = 1$ .
Solve for $p$ . Divide both sides by 3: $p = \frac{1}{3}$ .

Substituting back gives $(8^{\frac{1}{3}})^3 = 8$ . But the cube root also satisfies $(\sqrt[3]{8})^3 = 8$ . Since both expressions, when cubed, produce the same result, it follows that $8^{\frac{1}{3}} = \sqrt[3]{8}$ .

This derivation demonstrates how rational exponents arise naturally from the Power Property: the fractional exponent $\frac{1}{3}$ is the value that, when multiplied by $3$ (through the Power Property), yields $1$ . The same reasoning generalizes to any positive integer $n$ , establishing that $a^{\frac{1}{n}} = \sqrt[n]{a}$ .

Updated 2026-05-01

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