Here's the complete solution to all 20 problems, with explanations for each. The goal is to
simplify each square root expression by factoring out perfect squares.
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Key Concepts Used:
- √(a·b) = √a · √b (if a, b ≥ 0)
- √(a²) = |a|, but since we’re simplifying algebraic expressions and assuming variables represent non-negative numbers (standard in these problems), we can write √(x²) = x.
- For even exponents: √(x^{2k}) = x^k
- For odd exponents: √(x^{2k+1}) = x^k · √x
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##
✔ Solutions:
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1. √64
64 = 8² → √64 =
8
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2. –√18
18 = 9·2 = 3²·2 → √18 = 3√2 → –√18 =
–3√2
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3. √32
32 = 16·2 = 4²·2 → √32 =
4√2
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4. √50
50 = 25·2 = 5²·2 → √50 =
5√2
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5. √400
400 = 20² → √400 =
20
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6. √(x⁶)
x⁶ = (x³)² → √(x⁶) =
x³
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7. √(x⁷)
x⁷ = x⁶·x = (x³)²·x → √(x⁷) =
x³√x
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8. √(16x¹⁶)
16 = 4², x¹⁶ = (x⁸)² → √(16x¹⁶) =
4x⁸
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9. √(9x⁹)
9 = 3², x⁹ = x⁸·x = (x⁴)²·x → √(9x⁹) =
3x⁴√x
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10. √(40x⁸)
40 = 4·10 = 2²·10, x⁸ = (x⁴)² → √(40x⁸) =
2x⁴√10
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11. √(25x⁷)
25 = 5², x⁷ = x⁶·x = (x³)²·x → √(25x⁷) =
5x³√x
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12. √(12x⁵)
12 = 4·3 = 2²·3, x⁵ = x⁴·x = (x²)²·x → √(12x⁵) =
2x²√(3x)
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13. √(a²b⁴)
a² = (a)², b⁴ = (b²)² → √(a²b⁴) =
ab²
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14. √(49a⁸x¹²)
49 = 7², a⁸ = (a⁴)², x¹² = (x⁶)² → √(49a⁸x¹²) =
7a⁴x⁶
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15. √(28x⁹y⁶)
28 = 4·7 = 2²·7, x⁹ = x⁸·x = (x⁴)²·x, y⁶ = (y³)²
→ √(28x⁹y⁶) =
2x⁴y³√(7x)
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16. √(32m⁷n¹¹)
32 = 16·2 = 4²·2, m⁷ = m⁶·m = (m³)²·m, n¹¹ = n¹⁰·n = (n⁵)²·n
→ √(32m⁷n¹¹) =
4m³n⁵√(2mn)
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17. √(20x¹⁰y⁵)
20 = 4·5 = 2²·5, x¹⁰ = (x⁵)², y⁵ = y⁴·y = (y²)²·y
→ √(20x¹⁰y⁵) =
2x⁵y²√(5y)
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18. √(100ab⁴)
100 = 10², b⁴ = (b²)² → √(100ab⁴) =
10b²√a
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19. √(75x⁸y³)
75 = 25·3 = 5²·3, x⁸ = (x⁴)², y³ = y²·y → √(75x⁸y³) =
5x⁴y√(3y)
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20. √(98x⁷y⁵)
98 = 49·2 = 7²·2, x⁷ = x⁶·x = (x³)²·x, y⁵ = y⁴·y = (y²)²·y
→ √(98x⁷y⁵) =
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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✔ All simplified! Let me know if you’d like step-by-step breakdowns for any specific problem.
Parent Tip: Review the logic above to help your child master the concept of simplifying radicals with variables worksheet.
