Fractional Exponents Worksheet (printable, online, answers, examples) - Free Printable

Let’s solve each fractional exponent problem step by step.

Remember:
A fractional exponent like \( a^{\frac{m}{n}} \) means:
→ Take the n-th root of \( a \), then raise it to the m-th power.
OR
→ Raise \( a \) to the m-th power, then take the n-th root.
Both give the same result — pick whichever is easier!

We’ll go one by one.

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1. \( 36^{\frac{3}{2}} \)
Step 1: Square root of 36 → \( \sqrt{36} = 6 \)
Step 2: Cube that → \( 6^3 = 216 \)
✔ Answer: 216

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2. \( 32^{\frac{3}{5}} \)
Step 1: 5th root of 32 → \( \sqrt[5]{32} = 2 \) (because \( 2^5 = 32 \))
Step 2: Cube that → \( 2^3 = 8 \)
✔ Answer: 8

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3. \( 125^{\frac{2}{3}} \)
Step 1: Cube root of 125 → \( \sqrt[3]{125} = 5 \) (because \( 5^3 = 125 \))
Step 2: Square that → \( 5^2 = 25 \)
✔ Answer: 25

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4. \( 4^{\frac{5}{2}} \)
Step 1: Square root of 4 → \( \sqrt{4} = 2 \)
Step 2: Raise to 5th power → \( 2^5 = 32 \)
✔ Answer: 32

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5. \( 16^{\frac{3}{2}} \)
Step 1: Square root of 16 → \( \sqrt{16} = 4 \)
Step 2: Cube that → \( 4^3 = 64 \)
✔ Answer: 64

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6. \( 64^{\frac{2}{3}} \)
Step 1: Cube root of 64 → \( \sqrt[3]{64} = 4 \) (because \( 4^3 = 64 \))
Step 2: Square that → \( 4^2 = 16 \)
✔ Answer: 16

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Now the right column:

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7. \( 25^{\frac{3}{2}} \)
Step 1: Square root of 25 → \( \sqrt{25} = 5 \)
Step 2: Cube that → \( 5^3 = 125 \)
✔ Answer: 125

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8. \( 16^{\frac{5}{4}} \)
Step 1: 4th root of 16 → \( \sqrt[4]{16} = 2 \) (because \( 2^4 = 16 \))
Step 2: Raise to 5th power → \( 2^5 = 32 \)
✔ Answer: 32

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9. \( 81^{\frac{3}{4}} \)
Step 1: 4th root of 81 → \( \sqrt[4]{81} = 3 \) (because \( 3^4 = 81 \))
Step 2: Cube that → \( 3^3 = 27 \)
✔ Answer: 27

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10. \( 8^{\frac{4}{3}} \)
Step 1: Cube root of 8 → \( \sqrt[3]{8} = 2 \)
Step 2: Raise to 4th power → \( 2^4 = 16 \)
✔ Answer: 16

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11. \( 49^{\frac{1}{2}} \)
This is just square root of 49 → \( \sqrt{49} = 7 \)
✔ Answer: 7

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12. \( 27^{\frac{2}{3}} \)
Step 1: Cube root of 27 → \( \sqrt[3]{27} = 3 \)
Step 2: Square that → \( 3^2 = 9 \)
✔ Answer: 9

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Final Answer:
Left Column:
\( 36^{\frac{3}{2}} = 216 \)
\( 32^{\frac{3}{5}} = 8 \)
\( 125^{\frac{2}{3}} = 25 \)
\( 4^{\frac{5}{2}} = 32 \)
\( 16^{\frac{3}{2}} = 64 \)
\( 64^{\frac{2}{3}} = 16 \)

Right Column:
\( 25^{\frac{3}{2}} = 125 \)
\( 16^{\frac{5}{4}} = 32 \)
\( 81^{\frac{3}{4}} = 27 \)
\( 8^{\frac{4}{3}} = 16 \)
\( 49^{\frac{1}{2}} = 7 \)
\( 27^{\frac{2}{3}} = 9 \)