Rational Exponents

The Big Idea

Rational exponents are fractional exponents. They connect exponents and radicals into one unified system.

$x^{1/2} = \sqrt{x}$

A fractional exponent means “take a root.”

Think about the product rule: $x^a \cdot x^b = x^{a+b}$

What is $x^{1/2} \cdot x^{1/2}$ ?

\begin{aligned} x^{1/2} \cdot x^{1/2} &= x^{1/2 + 1/2} \\ &= x^1 \\ &= x \end{aligned}

So $x^{1/2}$ is something that, multiplied by itself, gives $x$ .

That’s exactly what $\sqrt{x}$ means.

$x^{1/n} = \sqrt[n]{x}$

The denominator tells you which root to take.

Examples:

What about $x^{m/n}$ ? There are two ways to compute it:

$x^{m/n} = \sqrt[n]{x^m} = \left(\sqrt[n]{x}\right)^m$

Take the root and raise to the power, in either order.

Tip: Root first is usually easier.

Example: Compute $8^{2/3}$

The exponent $\frac{2}{3}$ means: cube root, then square.

\begin{aligned} 8^{2/3} &= \left(\sqrt[3]{8}\right)^2 \\ &= 2^2 \\ &= 4 \end{aligned}

Example: Compute $27^{2/3}$

\begin{aligned} 27^{2/3} &= \left(\sqrt[3]{27}\right)^2 \\ &= 3^2 \\ &= 9 \end{aligned}

Example: Compute $16^{3/4}$

\begin{aligned} 16^{3/4} &= \left(\sqrt[4]{16}\right)^3 \\ &= 2^3 \\ &= 8 \end{aligned}

Combine two rules:

$x^{-m/n} = \frac{1}{x^{m/n}}$

Example: Compute $8^{-2/3}$

\begin{aligned} 8^{-2/3} &= \frac{1}{8^{2/3}} \\[0.5em] &= \frac{1}{4} \end{aligned}

Example: Compute $4^{-3/2}$

\begin{aligned} 4^{-3/2} &= \frac{1}{4^{3/2}} \\[0.5em] &= \frac{1}{(\sqrt{4})^3} \\[0.5em] &= \frac{1}{8} \end{aligned}

Rational exponents let you use exponent laws on radicals.

Without rational exponents:

$\sqrt{x} \cdot \sqrt[3]{x} = \text{ ??? }$

With rational exponents:

\begin{aligned} x^{1/2} \cdot x^{1/3} &= x^{1/2 + 1/3} \\ &= x^{5/6} \end{aligned}

The exponent laws make everything simpler.