Simplifying Radicals Worksheets - Math Monks - Free Printable

You're working on Simplifying Radicals: Perfect Square — this means you need to simplify square roots by factoring out perfect squares from inside the radical.

The key idea is:
> √(a²) = a (for non-negative a), and more generally, √(x·y) = √x · √y, so if you can factor out a perfect square, you can simplify.

We’ll go problem by problem.

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① √(25m²)

- 25 is 5², m² is (m)² → both perfect squares.
- √(25m²) = √25 · √m² = 5|m|
- *Note:* Since we’re simplifying radicals without context, we often assume variables represent non-negative numbers. So unless specified, we write 5m.

✔ Answer: 5m

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② √(121x¹⁶)

- 121 = 11², x¹⁶ = (x⁸)²
- √(121x¹⁶) = √121 · √x¹⁶ = 11 · x⁸ = 11x⁸

✔ Answer: 11x⁸

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③ √(9x²)

- 9 = 3², x² = (x)²
- √(9x²) = 3x

✔ Answer: 3x

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④ √(64w¹²)

- 64 = 8², w¹² = (w⁶)²
- √(64w¹²) = 8w⁶

✔ Answer: 8w⁶

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⑤ -√(100a⁴)

- 100 = 10², a⁴ = (a²)²
- √(100a⁴) = 10a² → with negative sign: -10a²

✔ Answer: -10a²

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⑥ √(a⁶b¹⁰)

- a⁶ = (a³)², b¹⁰ = (b⁵)²
- √(a⁶b¹⁰) = a³b⁵

✔ Answer: a³b⁵

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⑦ √(625y⁶⁴)

- 625 = 25² (or 5⁴), y⁶⁴ = (y³²)²
- √(625y⁶⁴) = 25y³²

✔ Answer: 25y³²

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⑧ √441

- 441 = 21² (since 20²=400, 21²=441)
- √441 = 21

✔ Answer: 21

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⑨ √(15a)

- 15 = 3×5 → no perfect square factors other than 1
- a has no exponent that’s even → cannot simplify further
- So it’s already simplified.

✔ Answer: √(15a)

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⑩ √(49y⁴m¹⁰⁰)

- 49 = 7², y⁴ = (y²)², m¹⁰⁰ = (m⁵⁰)²
- √(49y⁴m¹⁰⁰) = 7y²m⁵⁰

✔ Answer: 7y²m⁵⁰

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⑪ √(a²b⁴)

- a² = (a)², b⁴ = (b²)²
- √(a²b⁴) = ab²

✔ Answer: ab²

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⑫ √(16a²y⁴ / m¹⁶)

- Numerator: 16 = 4², a² = (a)², y⁴ = (y²)² → √(16a²y⁴) = 4ay²
- Denominator: m¹⁶ = (m⁸)² → √(m¹⁶) = m⁸
- So overall: 4ay² / m⁸

✔ Answer: (4ay²)/m⁸

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⑬ √(81/36)

- 81 = 9², 36 = 6²
- √(81/36) = √81 / √36 = 9/6 = 3/2

✔ Answer: 3/2

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⑭ √(x⁴ / y⁴¹⁰)

Wait — y⁴¹⁰? That seems like a typo. Probably meant y¹⁰ or y⁴?

Assuming it's y¹⁰ (most likely):

- x⁴ = (x²)², y¹⁰ = (y⁵)²
- √(x⁴/y¹⁰) = x² / y⁵

✔ Answer: x²/y⁵

*(If it really is y⁴¹⁰, then √(x⁴/y⁴¹⁰) = x² / y²⁰⁵ — but that’s extremely unlikely for this level.)*

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⑮ √(196a⁴)

- 196 = 14², a⁴ = (a²)²
- √(196a⁴) = 14a²

✔ Answer: 14a²

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⑯ √(729a¹²b¹⁶)

- 729 = 27² (or 9³, but 27² = 729), a¹² = (a⁶)², b¹⁶ = (b⁸)²
- √(729a¹²b¹⁶) = 27a⁶b⁸

✔ Answer: 27a⁶b⁸

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⑰ √(900/225)

- 900 = 30², 225 = 15²
- √(900/225) = 30/15 = 2

✔ Answer: 2

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⑱ √(361a⁴ / 121b⁴)

- 361 = 19², 121 = 11², a⁴ = (a²)², b⁴ = (b²)²
- √(361a⁴ / 121b⁴) = (19a²)/(11b²)

✔ Answer: (19a²)/(11b²)

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## ✔ Final Answers:

1. 5m
2. 11x⁸
3. 3x
4. 8w⁶
5. -10a²
6. a³b⁵
7. 25y³²
8. 21
9. √(15a)
10. 7y²m⁵⁰
11. ab²
12. (4ay²)/m⁸
13. 3/2
14. x²/y⁵ *(assuming typo: y¹⁰ instead of y⁴¹⁰)*
15. 14a²
16. 27a⁶b⁸
17. 2
18. (19a²)/(11b²)

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## 📝 Summary of Strategy:

To simplify radicals involving perfect squares:
1. Factor the number under the radical into perfect squares.
2. Use the property: √(a²b²) = ab (for non-negative a,b).
3. For fractions: √(a/b) = √a / √b
4. For variables: √(xⁿ) = x^(n/2) when n is even.
5. If no perfect square factors exist, leave as-is.

Let me know if you’d like to see step-by-step breakdowns for any specific problem!