Rational Exponents

Rational exponents are exponents written as fractions. The denominator is a root and the numerator is a power: a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m. For example, 8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 4. All the usual exponent rules still work with fraction exponents, so you can simplify radicals by switching to exponent form.

Before this lesson: You should know nth Roots, Exponent Rules, Zero and Negative Exponents and Adding and Subtracting Fractions.

Why a Fraction Exponent Means a Root

Exponent rules say (a^m)^n = a^{mn}. Suppose that rule still works for fractions. Then \left(9^{\frac{1}{2}}\right)^2 = 9^1 = 9. So 9^{\frac{1}{2}} is a number whose square is 9. That number is 3, the square root of 9.

The same idea works for any root. a^{\frac{1}{n}} is the nth root of a. A numerator other than 1 raises that root to a power.

The Rules

a^{\frac{1}{n}} = \sqrt[n]{a} \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m} \qquad a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}

Taking the root first usually keeps the numbers small. When a is negative, use these rules only with odd roots, like cube roots.

Worked Examples

Example 1: Unit fraction exponents

Evaluate (a) 25^{\frac{1}{2}}, (b) 27^{\frac{1}{3}} and (c) 16^{\frac{1}{4}}.

1. (a) Rewrite as a square root: 25^{\frac{1}{2}} = \sqrt{25}
2. Evaluate: 5
3. (b) Rewrite as a cube root (see nth roots): 27^{\frac{1}{3}} = \sqrt[3]{27}
4. Evaluate: 3
5. (c) Rewrite as a 4th root: 16^{\frac{1}{4}} = \sqrt[4]{16}
6. Evaluate: 2

Answer: (a) 5 (b) 3 (c) 2

Example 2: Numerators other than 1

Evaluate (a) 8^{\frac{2}{3}} and (b) 32^{\frac{3}{5}}.

1. (a) Write as a root, then a power: 8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2
2. Take the cube root: 2^2
3. Square: 4
4. (b) Write as a root, then a power: 32^{\frac{3}{5}} = \left(\sqrt[5]{32}\right)^3
5. Take the 5th root: 2^3
6. Cube: 8

Answer: (a) 4 (b) 8

Example 3: A negative rational exponent

Evaluate 81^{-\frac{3}{4}}.

1. A negative exponent means a reciprocal (see zero and negative exponents): 81^{-\frac{3}{4}} = \frac{1}{81^{\frac{3}{4}}}
2. Write the bottom as a root, then a power: \frac{1}{\left(\sqrt[4]{81}\right)^3}
3. Take the 4th root: \frac{1}{3^3}
4. Cube: \frac{1}{27}

Answer: \frac{1}{27}. The negative sign makes a reciprocal, not a negative number.

Example 4: Simplify with variables

Simplify (a) x^{\frac{1}{2}} \cdot x^{\frac{1}{3}} and (b) \left(16x^8\right)^{\frac{3}{4}}. Assume x is positive.

1. (a) Same base, so add the exponents (see exponent rules): x^{\frac{1}{2} + \frac{1}{3}}
2. Use a common denominator (see adding fractions): \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6}
3. Add: \frac{5}{6}, so the answer is x^{\frac{5}{6}}
4. (b) Apply the power to each factor: 16^{\frac{3}{4}} \cdot \left(x^8\right)^{\frac{3}{4}}
5. Write 16^{\frac{3}{4}} as a root, then a power: \left(\sqrt[4]{16}\right)^3 \cdot \left(x^8\right)^{\frac{3}{4}}
6. Take the 4th root: 2^3 \cdot \left(x^8\right)^{\frac{3}{4}}
7. Cube: 8 \cdot \left(x^8\right)^{\frac{3}{4}}
8. Multiply the exponents (see multiplying fractions): 8 \cdot \frac{3}{4} = 6, so 8x^6

Answer: (a) x^{\frac{5}{6}}, which is \sqrt[6]{x^5} (b) 8x^6

Example 5: Word problem with energy use

A common biology model, Kleiber’s law, says an animal’s resting energy use is about proportional to M^{\frac{3}{4}}, where M is its mass. How many times more energy does a 16 kg dog use at rest than a 1 kg rabbit?

1. Write the ratio of the two energy uses: \frac{16^{\frac{3}{4}}}{1^{\frac{3}{4}}}
2. Any power of 1 is 1: \frac{16^{\frac{3}{4}}}{1}
3. Write as a root, then a power: \left(\sqrt[4]{16}\right)^3
4. Take the 4th root: 2^3
5. Cube: 8

Answer: The dog uses about 8 times as much energy, even though it is 16 times as heavy.

Common Mistakes

Practice Problems

1. Evaluate 49^{\frac{1}{2}}.
2. Evaluate 125^{\frac{2}{3}}.
3. Evaluate 16^{-\frac{3}{2}}.
4. Evaluate (-27)^{\frac{1}{3}}.
5. Write \sqrt[5]{x^3} with a rational exponent.
6. Write y^{\frac{4}{3}} in radical form.
7. Simplify x^{\frac{2}{3}} \cdot x^{\frac{1}{6}}.
8. Simplify \left(27x^6\right)^{\frac{2}{3}}. Assume x is positive.
9. Simplify \frac{x^{\frac{3}{4}}}{x^{\frac{1}{2}}}.
10. Using Kleiber’s law (energy proportional to M^{\frac{3}{4}}), how many times more energy does an 81 kg animal use at rest than a 1 kg animal?

FAQ

What is a rational exponent?

It is an exponent that is a fraction, like \frac{2}{3}. The denominator tells you which root to take, and the numerator tells you which power to raise it to.

How do you convert a radical to a rational exponent?

Put the power over the index: \sqrt[n]{a^m} = a^{\frac{m}{n}}. For example, \sqrt[4]{x^3} = x^{\frac{3}{4}}.

What does a negative rational exponent mean?

It means take the reciprocal: a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}. So 8^{-\frac{1}{3}} = \frac{1}{2}.

Do the exponent rules work with fractions?

Yes. Add exponents when you multiply the same base, subtract when you divide, and multiply when you raise a power to a power. You just do fraction arithmetic on the exponents.

Summary

• a^{\frac{1}{n}} = \sqrt[n]{a}: the denominator is the root.
• a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m: the numerator is the power.
• A negative exponent means a reciprocal, not a negative number.
• All exponent rules work, using fraction arithmetic on the exponents.
• Take the root first to keep numbers small.

Previous lesson: nth Roots

Next lesson: Radical Functions

Back to the Algebra 2 Index