How To Find Cube Roots Worksheet, Definition, And Examples - Free Printable

Here's the solution to all 10 cube root problems from Mathcation.com’s “Cube Roots – Practice Worksheet A”, with explanations for each.

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🔢 What is a Cube Root?

The cube root of a number `x` is a number `y` such that:
> y³ = x

We write it as:
> ∛x = y

For example:
> ∛8 = 2, because 2 × 2 × 2 = 8

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## ✔ Solutions & Explanations:

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1. ∛729

We need to find a number that, when cubed, equals 729.

Try small integers:

- 5³ = 125
- 6³ = 216
- 7³ = 343
- 8³ = 512
- 9³ = 729 ← YES!

✔ Answer: 9

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2. ∛216

Try:

- 5³ = 125
- 6³ = 216 ← YES!

✔ Answer: 6

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3. ∛225

Check nearby cubes:

- 6³ = 216
- 7³ = 343 → too big

225 is between 216 and 343, so its cube root is not an integer.

Let’s approximate:

Since 225 - 216 = 9, and the gap to next cube is 343 - 216 = 127, we can estimate:

≈ 6 + (9/127) ≈ 6.07

But since this worksheet likely expects exact values or simplified radicals, let’s factor 225:

225 = 25 × 9 = 5² × 3² — no perfect cube factors.

So, ∛225 cannot be simplified further.

✔ Answer: ∛225 (already simplified)

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4. ∛200

Factor 200:

200 = 8 × 25 = 2³ × 5²

So,

∛200 = ∛(2³ × 5²) = 2 × ∛(5²) = 2∛25

✔ Answer: 2∛25

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5. ∛1728

Try:

- 10³ = 1000
- 11³ = 1331
- 12³ = 1728 ← YES!

✔ Answer: 12

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6. ∛343

Try:

- 6³ = 216
- 7³ = 343 ← YES!

✔ Answer: 7

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7. ∛10,000

Factor 10,000:

10,000 = 10⁴ = (2×5)⁴ = 2⁴ × 5⁴

We want to pull out perfect cubes:

2⁴ = 2³ × 2
5⁴ = 5³ × 5

So,

∛(2⁴ × 5⁴) = ∛(2³ × 5³ × 2 × 5) = 2×5 × ∛(10) = 10∛10

✔ Answer: 10∛10

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8. ∛577

Check cubes:

- 8³ = 512
- 9³ = 729 → too big

577 is between 512 and 729.

577 - 512 = 65
Gap to next cube: 729 - 512 = 217

Approximation: 8 + 65/217 ≈ 8.3

But 577 is prime? Let’s check divisibility:

577 ÷ 2, 3, 5, 7, 11, 13, 17, 19, 23 → none divide evenly.

So 577 has no perfect cube factors.

✔ Answer: ∛577 (cannot be simplified)

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9. ∛27,000

Factor 27,000:

27,000 = 27 × 1000 = 3³ × 10³ = (3×10)³ = 30³

So,

∛27,000 = ∛(30³) = 30

✔ Answer: 30

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10. ∛81

Factor 81:

81 = 3⁴ = 3³ × 3

So,

∛81 = ∛(3³ × 3) = 3 × ∛3 = 3∛3

✔ Answer: 3∛3

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## 📋 Final Answers Summary:

| Problem | Answer |
|---------|---------------|
| 1 | 9 |
| 2 | 6 |
| 3 | ∛225 |
| 4 | 2∛25 |
| 5 | 12 |
| 6 | 7 |
| 7 | 10∛10 |
| 8 | ∛577 |
| 9 | 30 |
| 10 | 3∛3 |

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## 💡 Tips for Solving Cube Roots:
- Memorize small cubes: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512, 9³=729, 10³=1000, 11³=1331, 12³=1728.
- Factor the number under the radical and look for groups of 3 identical prime factors.
- Pull out one factor for every group of 3, leave the rest inside.

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