You're asked to
simplify each square root expression. The goal is to factor out perfect squares from under the radical and simplify.
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🔑 Key Rules:
- √(a²) = a (for a ≥ 0)
- √(a·b) = √a · √b
- √(xⁿ) = x^(n/2) if n is even, or x^((n-1)/2)·√x if n is odd
- For variables: assume all variables represent non-negative real numbers unless otherwise specified.
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Let’s solve each one:
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1. √64
= √(8²) =
8
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2. -√18
= -√(9·2) = -√9·√2 = -3√2 →
-3√2
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3. √32
= √(16·2) = √16·√2 = 4√2 →
4√2
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4. √50
= √(25·2) = 5√2 →
5√2
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5. √400
= √(20²) =
20
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6. √(x⁶)
= √((x³)²) = |x³| → Since we usually assume x ≥ 0 in simplifying radicals, =
x³
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7. √(x⁷)
= √(x⁶·x) = √(x⁶)·√x = x³√x →
x³√x
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8. √(16x¹⁶)
= √16·√(x¹⁶) = 4·x⁸ →
4x⁸
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9. √(9x⁹)
= √9·√(x⁸·x) = 3·x⁴·√x →
3x⁴√x
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10. √(40x⁸)
= √(4·10·x⁸) = √4·√10·√(x⁸) = 2·√10·x⁴ →
2x⁴√10
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11. √(25x⁷)
= √25·√(x⁶·x) = 5·x³·√x →
5x³√x
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12. √(12x⁵)
= √(4·3·x⁴·x) = √4·√3·√(x⁴)·√x = 2·√3·x²·√x =
2x²√(3x)
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13. √(a²b⁴)
= √(a²)·√(b⁴) = |a|·b² → Assuming a ≥ 0 →
ab²
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14. √(49a⁸x¹²)
= √49·√(a⁸)·√(x¹²) = 7·a⁴·x⁶ →
7a⁴x⁶
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15. √(28x⁹y⁶)
= √(4·7·x⁸·x·y⁶) = √4·√7·√(x⁸)·√x·√(y⁶)
= 2·√7·x⁴·√x·y³ =
2x⁴y³√(7x)
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16. √(32m⁷n¹¹)
= √(16·2·m⁶·m·n¹⁰·n)
= √16·√2·√(m⁶)·√m·√(n¹⁰)·√n
= 4·√2·m³·√m·n⁵·√n =
4m³n⁵√(2mn)
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17. √(20x¹⁰y⁵)
= √(4·5·x¹⁰·y⁴·y)
= √4·√5·√(x¹⁰)·√(y⁴)·√y
= 2·√5·x⁵·y²·√y =
2x⁵y²√(5y)
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18. √(100ab⁴)
= √100·√a·√(b⁴) = 10·√a·b² =
10b²√a
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19. √(75x⁸y³)
= √(25·3·x⁸·y²·y)
= √25·√3·√(x⁸)·√(y²)·√y
= 5·√3·x⁴·y·√y =
5x⁴y√(3y)
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20. √(98x⁷y⁵)
= √(49·2·x⁶·x·y⁴·y)
= √49·√2·√(x⁶)·√x·√(y⁴)·√y
= 7·√2·x³·√x·y²·√y =
7x³y²√(2xy)
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##
✔ Final Answers:
1.
8
2.
-3√2
3.
4√2
4.
5√2
5.
20
6.
x³
7.
x³√x
8.
4x⁸
9.
3x⁴√x
10.
2x⁴√10
11.
5x³√x
12.
2x²√(3x)
13.
ab²
14.
7a⁴x⁶
15.
2x⁴y³√(7x)
16.
4m³n⁵√(2mn)
17.
2x⁵y²√(5y)
18.
10b²√a
19.
5x⁴y√(3y)
20.
7x³y²√(2xy)
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Let me know if you’d like to see step-by-step breakdowns for any specific problem!
Parent Tip: Review the logic above to help your child master the concept of simplifying radicals worksheet algebra 1.
