Unit Fraction Exponents Worksheet (printable, online, answers ... - Free Printable

You're working with unit fraction exponents, which means the exponent is of the form `1/n` — this represents the n-th root of the base.

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🔢 Key Concept:

> For any positive real number `a` and positive integer `n`,
> `a^(1/n) = ⁿ√a` (the n-th root of a)

So:

- `a^(1/2)` = √a (square root)
- `a^(1/3)` = ∛a (cube root)
- `a^(1/4)` = ⁴√a (fourth root)
- etc.

We’ll evaluate each expression by finding the appropriate root.

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## ✔ Left Column Solutions:

1. `25^(1/2)`

= √25 = 5

2. `81^(1/4)`

= ⁴√81
Since 81 = 3⁴ → ⁴√(3⁴) = 3

3. `64^(1/2)`

= √64 = 8

4. `4^(1/2)`

= √4 = 2

5. `125^(1/3)`

= ∛125
Since 125 = 5³ → ∛(5³) = 5

6. `32^(1/5)`

= ⁵√32
Since 32 = 2⁵ → ⁵√(2⁵) = 2

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## ✔ Right Column Solutions:

1. `36^(1/2)`

= √36 = 6

2. `16^(1/4)`

= ⁴√16
Since 16 = 2⁴ → ⁴√(2⁴) = 2

3. `81^(1/4)`

= ⁴√81 = 3 (same as above — repeated)

4. `8^(1/3)`

= ∛8
Since 8 = 2³ → ∛(2³) = 2

5. `49^(1/2)`

= √49 = 7

6. `27^(1/3)`

= ∛27
Since 27 = 3³ → ∛(3³) = 3

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## 📝 Final Answers (in order):

Left Column:
- 25^(1/2) = 5
- 81^(1/4) = 3
- 64^(1/2) = 8
- 4^(1/2) = 2
- 125^(1/3) = 5
- 32^(1/5) = 2

Right Column:
- 36^(1/2) = 6
- 16^(1/4) = 2
- 81^(1/4) = 3
- 8^(1/3) = 2
- 49^(1/2) = 7
- 27^(1/3) = 3

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✔ All answers are positive real numbers, since we’re dealing with principal roots (standard for these problems unless otherwise specified).

Let me know if you want to see how to compute these without knowing perfect powers — or how to handle negative bases!