1Cademy - Simplifying $$\left(\frac{3}{7}\right)^2$$, $$\left(\frac{b}{3}\right)^4$$, and $$\left(\frac{k}{j}\right)^3$$ Using the Quotient to a Power Property

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Quotient to a Power Property for Exponents

Example

Simplifying $\left(\frac{3}{7}\right)^2$ , $\left(\frac{b}{3}\right)^4$ , and $\left(\frac{k}{j}\right)^3$ Using the Quotient to a Power Property

Apply the Quotient to a Power Property to simplify three expressions in which a fraction is raised to a power — one with purely numerical terms, one with a variable numerator and numerical denominator, and one with two variables.

ⓐ $\left(\frac{3}{7}\right)^2 = \frac{9}{49}$ : The base is the fraction $\frac{3}{7}$ and the exponent is $2$ . Use the Quotient to a Power Property to raise the numerator and denominator each to the second power: $\left(\frac{3}{7}\right)^2 = \frac{3^2}{7^2}$ . Evaluate: $3^2 = 9$ and $7^2 = 49$ , giving $\frac{9}{49}$ .

ⓑ $\left(\frac{b}{3}\right)^4 = \frac{b^4}{81}$ : The base is the fraction $\frac{b}{3}$ and the exponent is $4$ . Apply the property: $\left(\frac{b}{3}\right)^4 = \frac{b^4}{3^4}$ . Evaluate the denominator: $3^4 = 81$ , giving $\frac{b^4}{81}$ .

ⓒ $\left(\frac{k}{j}\right)^3 = \frac{k^3}{j^3}$ : The base is the fraction $\frac{k}{j}$ and the exponent is $3$ . Raise both the numerator and denominator to the third power: $\left(\frac{k}{j}\right)^3 = \frac{k^3}{j^3}$ .

Parts (a) and (b) show that when the numerator or denominator is a number, the resulting power can be computed to produce a single numerical value. Part (c) shows that when both the numerator and denominator are variables, the simplified form retains each variable raised to the exponent.

Updated 2026-04-29

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