10. Powers and roots of fractions — Fractions

Power of a fraction

Raising a fraction to the power $n$ means multiplying it by itself $n$ times. Both numerator and denominator are raised:

\left(\dfrac{a}{b}\right)^n = \underbrace{\dfrac{a}{b} \cdot \dfrac{a}{b} \cdots \dfrac{a}{b}}_{n} = \dfrac{a^n}{b^n}

Example.

\left(\dfrac{3}{2}\right)^4 = \dfrac{3}{2} \cdot \dfrac{3}{2} \cdot \dfrac{3}{2} \cdot \dfrac{3}{2} = \dfrac{3^4}{2^4} = \dfrac{81}{16}

Operations with powers of fractions

Same base

To multiply (or divide) powers with the same base, keep the base and add (or subtract) the exponents.

\left(\dfrac{a}{b}\right)^m \cdot \left(\dfrac{a}{b}\right)^n = \left(\dfrac{a}{b}\right)^{m+n}

\left(\dfrac{a}{b}\right)^m : \left(\dfrac{a}{b}\right)^n = \left(\dfrac{a}{b}\right)^{m-n}

Same exponent

To multiply (or divide) powers with the same exponent, keep the exponent and multiply (or divide) the bases.

\left(\dfrac{a}{b}\right)^m \cdot \left(\dfrac{c}{d}\right)^m = \left(\dfrac{a}{b} \cdot \dfrac{c}{d}\right)^m

\left(\dfrac{a}{b}\right)^m : \left(\dfrac{c}{d}\right)^m = \left(\dfrac{a}{b} : \dfrac{c}{d}\right)^m

Power of a power

Keep the base and multiply the exponents.

\left[\left(\dfrac{a}{b}\right)^m\right]^n = \left(\dfrac{a}{b}\right)^{m \cdot n}

Negative exponent

It is the reciprocal of the fraction, with the sign of the exponent changed.

\left(\dfrac{a}{b}\right)^{-m} = \left(\dfrac{b}{a}\right)^m

Exponent 0

Anything raised to the power $0$ equals $1$.

\left(\dfrac{a}{b}\right)^0 = 1

Exponent 1

Anything raised to the power $1$ is itself.

\left(\dfrac{a}{b}\right)^1 = \dfrac{a}{b}

Examples

Example 1. Flip $\dfrac{4}{3}$ to get the same base as $\dfrac{3}{4}$:

\left(\dfrac{3}{4}\right)^5 \cdot \left(\dfrac{4}{3}\right)^2 \cdot \left(\dfrac{1}{3}\right)^3 = \left(\dfrac{3}{4}\right)^5 \cdot \left(\dfrac{3}{4}\right)^{-2} \cdot \left(\dfrac{1}{3}\right)^3

Same base: add the exponents, $5 + (-2) = 3$. Now both have the same exponent, so multiply the bases.

= \left(\dfrac{3}{4}\right)^3 \cdot \left(\dfrac{1}{3}\right)^3 = \left(\dfrac{3}{4} \cdot \dfrac{1}{3}\right)^3 = \left(\dfrac{1}{4}\right)^3 = \dfrac{1}{64}

Example 2. The square bracket first, from the inside out:

\left[\left(\dfrac{2}{3}\right)^7 : \left(\dfrac{2}{3}\right)^3\right]^2 : \left(\dfrac{3}{2}\right)^8 = \left[\left(\dfrac{2}{3}\right)^4\right]^2 : \left(\dfrac{3}{2}\right)^8 = \left(\dfrac{2}{3}\right)^8 : \left(\dfrac{3}{2}\right)^8

Same exponent: divide the bases.

= \left(\dfrac{2}{3} : \dfrac{3}{2}\right)^8 = \left(\dfrac{4}{9}\right)^8 = \dfrac{4^8}{9^8}

Another route: $\dfrac{3}{2} = \left(\dfrac{2}{3}\right)^{-1}$, so dividing by $\left(\dfrac{3}{2}\right)^8$ is multiplying by $\left(\dfrac{2}{3}\right)^8$, which gives $\left(\dfrac{2}{3}\right)^{16}$. Same number: $\dfrac{4^8}{9^8} = \dfrac{2^{16}}{3^{16}}$.

Root of a fraction

Root

The root of a fraction is the root of the numerator over the root of the denominator.

\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}

Example.

\sqrt{\dfrac{49}{25}} = \dfrac{\sqrt{49}}{\sqrt{25}} = \dfrac{7}{5}