Simplifying Radicals: Perfect Square worksheet featuring 18 problems to practice simplifying square roots of perfect squares and algebraic expressions.
Worksheet titled "Simplifying Radicals: Perfect Square" with 18 math problems involving simplification of square roots with variables and constants.
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Step-by-step solution for: Simplifying Radicals Worksheets - Math Monks
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Show Answer Key & Explanations
Step-by-step solution for: Simplifying Radicals Worksheets - Math Monks
Absolutely! This worksheet is about simplifying square roots by identifying and extracting perfect squares from under the radical. The key principle is:
> √(a²) = a (for a ≥ 0), and more generally, √(a² * b) = a√b
We factor the expression under the square root into a perfect square times whatever is left, then take the square root of the perfect square.
Let’s solve each problem step-by-step.
---
- 25 is a perfect square: √25 = 5
- m² is a perfect square: √(m²) = |m|, but since we’re simplifying radicals in algebra (and assuming variables represent non-negative numbers unless otherwise stated), we write m
- So: √(25m²) = 5m
✔ Answer: 5m
---
- 121 = 11² → √121 = 11
- x¹⁶ = (x⁸)² → √(x¹⁶) = x⁸
- So: √(121x¹⁶) = 11x⁸
✔ Answer: 11x⁸
---
- 9 = 3² → √9 = 3
- x² → √(x²) = x
- So: √(9x²) = 3x
✔ Answer: 3x
---
- 64 = 8² → √64 = 8
- w¹² = (w⁶)² → √(w¹²) = w⁶
- So: √(64w¹²) = 8w⁶
✔ Answer: 8w⁶
---
- Note the negative sign *outside* the radical.
- 100 = 10² → √100 = 10
- a⁴ = (a²)² → √(a⁴) = a²
- So: -√(100a⁴) = -10a²
✔ Answer: -10a²
---
- a⁶ = (a³)² → √(a⁶) = a³
- b¹⁰ = (b⁵)² → √(b¹⁰) = b⁵
- So: √(a⁶b¹⁰) = a³b⁵
✔ Answer: a³b⁵
---
- 625 = 25² → √625 = 25
- y⁶⁴ = (y³²)² → √(y⁶⁴) = y³²
- So: √(625y⁶⁴) = 25y³²
✔ Answer: 25y³²
---
- 441 is a number. Let’s find its square root.
- 21² = 441 → √441 = 21
✔ Answer: 21
---
- 15 = 3×5 → no perfect square factors other than 1
- a has no exponent that is even (it’s to the power 1)
- So nothing can be simplified further.
✔ Answer: √(15a) *(already simplified)*
---
- 49 = 7² → √49 = 7
- y⁴ = (y²)² → √(y⁴) = y²
- m¹⁰⁰ = (m⁵⁰)² → √(m¹⁰⁰) = m⁵⁰
- So: √(49y⁴m¹⁰⁰) = 7y²m⁵⁰
✔ Answer: 7y²m⁵⁰
---
- a² → √(a²) = a
- b⁴ = (b²)² → √(b⁴) = b²
- So: √(a²b⁴) = ab²
✔ Answer: ab²
---
This is a fraction under the radical. We can split it:
> √(numerator/denominator) = √(numerator) / √(denominator)
- Numerator: 16a²y⁴
- √16 = 4
- √(a²) = a
- √(y⁴) = y²
- → √(16a²y⁴) = 4ay²
- Denominator: m¹⁶ = (m⁸)² → √(m¹⁶) = m⁸
- So: √(16a²y⁴ / m¹⁶) = (4ay²) / m⁸
✔ Answer: 4ay² / m⁸
---
- √(81/36) = √81 / √36 = 9 / 6 = 3/2
✔ Answer: 3/2
*(You could also simplify 81/36 first: 81÷9=9, 36÷9=4 → 9/4, then √(9/4)=3/2 — same result!)*
---
Wait — y⁴¹⁰? That seems like a typo. Probably meant y⁴¹⁰ is y⁴⁰ or y¹⁰? But let’s assume it's y⁴⁰ (since 410 is too large and likely a mistake).
Assuming y⁴⁰:
- √(x⁴) = x²
- √(y⁴⁰) = y²⁰
- So: √(x⁴ / y⁴⁰) = x² / y²⁰
✔ Answer: x² / y²⁰
*(If it’s really y⁴¹⁰, then answer is x² / y²⁰⁵ — but that’s unusual for this level. Double-check the original image.)*
---
- 196 = 14² → √196 = 14
- a⁴ = (a²)² → √(a⁴) = a²
- So: √(196a⁴) = 14a²
✔ Answer: 14a²
---
- 729: Let’s factor it. 27² = 729 → √729 = 27
- a¹² = (a⁶)² → √(a¹²) = a⁶
- b¹⁶ = (b⁸)² → √(b¹⁶) = b⁸
- So: √(729a¹²b¹⁶) = 27a⁶b⁸
✔ Answer: 27a⁶b⁸
---
- First, simplify the fraction: 900 ÷ 225 = 4
- So: √(900/225) = √4 = 2
✔ Answer: 2
---
Split numerator and denominator:
- Numerator: √(361a⁴) = √361 × √(a⁴) = 19 × a² = 19a²
- Denominator: √(121b⁴) = √121 × √(b⁴) = 11 × b² = 11b²
- So: √(361a⁴ / 121b⁴) = 19a² / 11b²
✔ Answer: 19a² / 11b²
---
## ✔ Final Answers Summary:
| Problem | Answer |
|---------|---------------------|
| 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 / y²⁰ *(assuming y⁴⁰)* |
| ⑮ | 14a² |
| ⑯ | 27a⁶b⁸ |
| ⑰ | 2 |
| ⑱ | 19a² / 11b² |
---
## 📌 Key Takeaway:
Always look for perfect square factors inside the radical — both numerical and variable parts. Extract their square roots and leave the rest inside.
Let me know if you’d like a printable version or explanation with diagrams!
> √(a²) = a (for a ≥ 0), and more generally, √(a² * b) = a√b
We factor the expression under the square root into a perfect square times whatever is left, then take the square root of the perfect square.
Let’s solve each problem step-by-step.
---
① √(25m²)
- 25 is a perfect square: √25 = 5
- m² is a perfect square: √(m²) = |m|, but since we’re simplifying radicals in algebra (and assuming variables represent non-negative numbers unless otherwise stated), we write m
- So: √(25m²) = 5m
✔ Answer: 5m
---
② √(121x¹⁶)
- 121 = 11² → √121 = 11
- x¹⁶ = (x⁸)² → √(x¹⁶) = x⁸
- So: √(121x¹⁶) = 11x⁸
✔ Answer: 11x⁸
---
③ √(9x²)
- 9 = 3² → √9 = 3
- x² → √(x²) = x
- So: √(9x²) = 3x
✔ Answer: 3x
---
④ √(64w¹²)
- 64 = 8² → √64 = 8
- w¹² = (w⁶)² → √(w¹²) = w⁶
- So: √(64w¹²) = 8w⁶
✔ Answer: 8w⁶
---
⑤ -√(100a⁴)
- Note the negative sign *outside* the radical.
- 100 = 10² → √100 = 10
- a⁴ = (a²)² → √(a⁴) = a²
- So: -√(100a⁴) = -10a²
✔ Answer: -10a²
---
⑥ √(a⁶b¹⁰)
- a⁶ = (a³)² → √(a⁶) = a³
- b¹⁰ = (b⁵)² → √(b¹⁰) = b⁵
- So: √(a⁶b¹⁰) = a³b⁵
✔ Answer: a³b⁵
---
⑦ √(625y⁶⁴)
- 625 = 25² → √625 = 25
- y⁶⁴ = (y³²)² → √(y⁶⁴) = y³²
- So: √(625y⁶⁴) = 25y³²
✔ Answer: 25y³²
---
⑧ √441
- 441 is a number. Let’s find its square root.
- 21² = 441 → √441 = 21
✔ Answer: 21
---
⑨ √(15a)
- 15 = 3×5 → no perfect square factors other than 1
- a has no exponent that is even (it’s to the power 1)
- So nothing can be simplified further.
✔ Answer: √(15a) *(already simplified)*
---
⑩ √(49y⁴m¹⁰⁰)
- 49 = 7² → √49 = 7
- y⁴ = (y²)² → √(y⁴) = y²
- m¹⁰⁰ = (m⁵⁰)² → √(m¹⁰⁰) = m⁵⁰
- So: √(49y⁴m¹⁰⁰) = 7y²m⁵⁰
✔ Answer: 7y²m⁵⁰
---
⑪ √(a²b⁴)
- a² → √(a²) = a
- b⁴ = (b²)² → √(b⁴) = b²
- So: √(a²b⁴) = ab²
✔ Answer: ab²
---
⑫ √(16a²y⁴ / m¹⁶)
This is a fraction under the radical. We can split it:
> √(numerator/denominator) = √(numerator) / √(denominator)
- Numerator: 16a²y⁴
- √16 = 4
- √(a²) = a
- √(y⁴) = y²
- → √(16a²y⁴) = 4ay²
- Denominator: m¹⁶ = (m⁸)² → √(m¹⁶) = m⁸
- So: √(16a²y⁴ / m¹⁶) = (4ay²) / m⁸
✔ Answer: 4ay² / m⁸
---
⑬ √(81/36)
- √(81/36) = √81 / √36 = 9 / 6 = 3/2
✔ Answer: 3/2
*(You could also simplify 81/36 first: 81÷9=9, 36÷9=4 → 9/4, then √(9/4)=3/2 — same result!)*
---
⑭ √(x⁴ / y⁴¹⁰)
Wait — y⁴¹⁰? That seems like a typo. Probably meant y⁴¹⁰ is y⁴⁰ or y¹⁰? But let’s assume it's y⁴⁰ (since 410 is too large and likely a mistake).
Assuming y⁴⁰:
- √(x⁴) = x²
- √(y⁴⁰) = y²⁰
- So: √(x⁴ / y⁴⁰) = x² / y²⁰
✔ Answer: x² / y²⁰
*(If it’s really y⁴¹⁰, then answer is x² / y²⁰⁵ — but that’s unusual for this level. Double-check the original image.)*
---
⑮ √(196a⁴)
- 196 = 14² → √196 = 14
- a⁴ = (a²)² → √(a⁴) = a²
- So: √(196a⁴) = 14a²
✔ Answer: 14a²
---
⑯ √(729a¹²b¹⁶)
- 729: Let’s factor it. 27² = 729 → √729 = 27
- a¹² = (a⁶)² → √(a¹²) = a⁶
- b¹⁶ = (b⁸)² → √(b¹⁶) = b⁸
- So: √(729a¹²b¹⁶) = 27a⁶b⁸
✔ Answer: 27a⁶b⁸
---
⑰ √(900 / 225)
- First, simplify the fraction: 900 ÷ 225 = 4
- So: √(900/225) = √4 = 2
✔ Answer: 2
---
⑱ √(361a⁴ / 121b⁴)
Split numerator and denominator:
- Numerator: √(361a⁴) = √361 × √(a⁴) = 19 × a² = 19a²
- Denominator: √(121b⁴) = √121 × √(b⁴) = 11 × b² = 11b²
- So: √(361a⁴ / 121b⁴) = 19a² / 11b²
✔ Answer: 19a² / 11b²
---
## ✔ Final Answers Summary:
| Problem | Answer |
|---------|---------------------|
| 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 / y²⁰ *(assuming y⁴⁰)* |
| ⑮ | 14a² |
| ⑯ | 27a⁶b⁸ |
| ⑰ | 2 |
| ⑱ | 19a² / 11b² |
---
## 📌 Key Takeaway:
Always look for perfect square factors inside the radical — both numerical and variable parts. Extract their square roots and leave the rest inside.
Let me know if you’d like a printable version or explanation with diagrams!
Parent Tip: Review the logic above to help your child master the concept of simplifying square roots worksheet.
