Lesson 9.8: Rational Exponents

New Concept

Rational exponents are a powerful way to express radicals, like writing $\sqrt[n]{a}$ as $a^{\frac{1}{n}}$. This lets you use familiar exponent rules to simplify complex expressions involving roots, making them much easier to work with.

What’s next

Next, you'll dive into practice cards and interactive examples to master simplifying expressions using these new fractional exponents.

Property

If $\sqrt[n]{a}$ is a real number and $n \ge 2$, $a^{\frac{1}{n}} = \sqrt[n]{a}$.
Rational exponents are another way of writing expressions with radicals.

Examples

• To write $b^{\frac{1}{5}}$ as a radical, the denominator 5 becomes the index of the root, so we get $\sqrt[5]{b}$.

• To simplify $81^{\frac{1}{4}}$, we rewrite it as $\sqrt[4]{81}$. Since $3^4 = 81$, the answer is $3$.

Property

For any positive integers $m$ and $n$,
$a^{\frac{m}{n}} = (a^{\frac{1}{n}})^m = (\sqrt[n]{a})^m$
and
$a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}$
To simplify, it is usually best to take the root first to keep the numbers in the radicand smaller.

Examples

• To simplify $27^{\frac{2}{3}}$, we can write it as $(\sqrt[3]{27})^2$. The cube root of 27 is 3, and $3^2$ is $9$.

• To simplify $16^{\frac{5}{4}}$, we write it as $(\sqrt[4]{16})^5$. The fourth root of 16 is 2, and $2^5$ is $32$.

Property

If $a, b$ are real numbers and $m, n$ are rational numbers, then

• Product Property $a^m \cdot a^n = a^{m+n}$
• Power Property $(a^m)^n = a^{mn}$
• Product to a Power $(ab)^m = a^m b^m$
• Quotient Property $\frac{a^m}{a^n} = a^{m-n}$, $a \ne 0$
• Zero Exponent Definition $a^0 = 1$, $a \ne 0$
• Quotient to a Power Property $(\frac{a}{b})^m = \frac{a^m}{b^m}$, $b \ne 0$
• Negative Exponent Property $a^{-n} = \frac{1}{a^n}$, $a \ne 0$

Examples

• Using the Product Property: $x^{\frac{1}{3}} \cdot x^{\frac{5}{3}} = x^{\frac{1}{3} + \frac{5}{3}} = x^{\frac{6}{3}} = x^2$.

• Using the Power Property: $(y^{10})^{\frac{2}{5}} = y^{10 \cdot \frac{2}{5}} = y^{\frac{20}{5}} = y^4$.