Rational Exponents

By the end of this lesson, you will be able to convert between radical and rational exponent notation, apply exponent rules to expressions with rational exponents, and simplify expressions involving both.

The Big Idea

Rational exponents unify the notation for roots and powers into a single system. Instead of writing $\sqrt[n]{x}$ for roots, we write $x^{1/n}$. This makes applying exponent rules to radicals straightforward — you already know how to multiply and divide exponents; now you just need to allow fractions.

Definition of Rational Exponents

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

$$x^{m/n} = \left(x^{1/n}\right)^m = \left(\sqrt[n]{x}\right)^m = \sqrt[n]{x^m}$$

Both interpretations are valid: $x^{m/n} = \left(\sqrt[n]{x}\right)^m = \sqrt[n]{x^m}$.

For even $n$: $x \geq 0$ (no even root of a negative real number)

For odd $n$: any real $x$ is allowed.

Exponent Rules with Rational Exponents

All standard exponent rules apply:

Rule	Example
$x^a \cdot x^b = x^{a+b}$	$x^{1/2} \cdot x^{1/3} = x^{5/6}$
$(x^a)^b = x^{ab}$	$(x^{2/3})^3 = x^2$
$(xy)^a = x^a y^a$	$(4x)^{1/2} = 2x^{1/2}$
$x^{-a} = \dfrac{1}{x^a}$	$x^{-3/4} = \dfrac{1}{x^{3/4}}$
$\dfrac{x^a}{x^b} = x^{a-b}$	$\dfrac{x^{3/2}}{x^{1/2}} = x$

Example 1 — Converting Notation

Write as a radical: $x^{3/4} = \sqrt[4]{x^3}$

Write as a rational exponent: $\sqrt[5]{x^2} = x^{2/5}$

Evaluate: $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$

Example 2 — Simplifying Expressions

(a) $\left(\dfrac{x^2}{y^4}\right)^{3/2} = \dfrac{x^3}{y^6}$

(b) $\dfrac{x^{5/3}}{x^{2/3}} = x^{5/3 - 2/3} = x$

(c) $(8x^6)^{2/3} = 8^{2/3} \cdot x^4 = 4x^4$

Example 3 — Negative Rational Exponents

$$\left(\frac{27}{8}\right)^{-2/3} = \left(\frac{8}{27}\right)^{2/3} = \left(\sqrt[3]{\frac{8}{27}}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$$

Example 4 — Combining with Algebra

Simplify: $\dfrac{(x^{1/2} + y^{1/2})(x^{1/2} - y^{1/2})}{x - y}$

Numerator: $x^{1/2 \cdot 2} - y^{1/2 \cdot 2} = x - y$ (difference of squares with $a = x^{1/2}$, $b = y^{1/2}$)