Rational Exponents
Rational Exponents
Rational exponents provide an alternative notation for radicals. The expression a^(1/n) means the nth root of a, and a^(m/n) means the nth root of a raised to the mth power. This notation is more compact and makes the laws of exponents directly applicable to roots.
Because rational exponents and radicals represent the same mathematical objects, you can convert freely between the two forms, choosing whichever is more convenient for the problem at hand.
Rational exponents provide an alternative notation for radicals and extend the exponent rules to fractional powers.
For $a \ge 0$ and positive integer $n$:
$$a^{1/n} = \sqrt[n]{a}$$
More generally:
$$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$
All integer exponent laws extend to rational exponents:
| Law | Rule |
|---|---|
| Product | $a^m \cdot a^n = a^{m+n}$ |
| Quotient | $\dfrac{a^m}{a^n} = a^{m-n}$ |
| Power of a power | $(a^m)^n = a^{mn}$ |
| Power of a product | $(ab)^n = a^n b^n$ |
| Negative exponent | $a^{-n} = \dfrac{1}{a^n}$ |
Evaluate $8^{2/3}$.
$$8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$$
Evaluate $27^{-1/3}$.
$$27^{-1/3} = \frac{1}{27^{1/3}} = \frac{1}{\sqrt[3]{27}} = \frac{1}{3}$$
Rewrite $\sqrt[4]{x^3}$ using rational exponents.
$$\sqrt[4]{x^3} = x^{3/4}$$
Simplify $x^{1/2} \cdot x^{2/3}$.
$$x^{1/2 + 2/3} = x^{3/6 + 4/6} = x^{7/6}$$
Simplify $\dfrac{a^{5/6}}{a^{1/3}}$.
$$a^{5/6 - 1/3} = a^{5/6 - 2/6} = a^{3/6} = a^{1/2} = \sqrt{a}$$
Evaluate $\left(\dfrac{16}{81}\right)^{3/4}$.
$$\left(\frac{16}{81}\right)^{3/4} = \frac{16^{3/4}}{81^{3/4}} = \frac{(\sqrt[4]{16})^3}{(\sqrt[4]{81})^3} = \frac{2^3}{3^3} = \frac{8}{27}$$
Practice Problems
Show Answer Key
1. $2$
2. $(\sqrt[5]{32})^2 = 2^2 = 4$
3. $\dfrac{1}{125^{2/3}} = \dfrac{1}{25}$
4. $x^{3/5}$
5. $\dfrac{1}{\sqrt[4]{y^3}}$
6. $x^{5/6}$
7. $a^{1/4}$
8. $x^2$
9. $\dfrac{4}{9}$
10. $\dfrac{1}{7}$
11. $x^0 = 1$
12. $(\sqrt[6]{64})^5 = 2^5 = 32$
13. $8a^3$
14. $1000$
15. $x^3 y^2$