Fractional Exponents Practice Worksheet

Let’s solve each problem step by step.

We’re working with fractional exponents and radicals. The key rule is:

> x^(a/b) = the b-th root of x^a, or written as √[b](x^a)

Also, if there’s no number on top of the radical (like √x), it means square root — so that’s the same as exponent 1/2.

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Part 1: Write in radical form

This means turn expressions like 5^(1/3) into roots.

Rule:
x^(m/n) = √[n](x^m)

So:

1.) 5^(1/3) → cube root of 5 → √[3]{5}

2.) 6^(4/5) → fifth root of 6^4 → √[5]{6⁴}

3.) 2^(3/2) → square root of 2³ → √{2³} (since denominator 2 means square root)

4.) 9^(1/4) → fourth root of 9 → √[4]{9}

5.) 3^(3/4) → fourth root of 3³ → √[4]{3³}

6.) 6^(1/5) → fifth root of 6 → √[5]{6}

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Part 2: Write in exponential form

This means turn radicals into fractions in the exponent.

Rule:
√[n]{x^m} = x^(m/n)

If it’s just √x, that’s x^(1/2). If it’s ∛x, that’s x^(1/3), etc.

7.) √10 → this is square root → 10^(1/2)

8.) √(5³) → square root of 5 cubed → 5^(3/2)

9.) ∛3 → cube root of 3 → 3^(1/3)

10.) ∜(2³) → fourth root of 2 cubed → 2^(3/4)

11.) ∛(5²) → cube root of 5 squared → 5^(2/3)

12.) ∛7 → cube root of 7 → 7^(1/3)

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Now let’s write all answers clearly.

Final Answer:
1.) √[3]{5}
2.) √[5]{6⁴}
3.) √{2³}
4.) √[4]{9}
5.) √[4]{3³}
6.) √[5]{6}
7.) 10^(1/2)
8.) 5^(3/2)
9.) 3^(1/3)
10.) 2^(3/4)
11.) 5^(2/3)
12.) 7^(1/3)