Here's the complete solution to the “Multiplying Radicals” worksheet, with step-by-step explanations for each problem.
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Key Rule:
When multiplying square roots:
√a · √b = √(a·b) — you multiply the numbers inside the radicals.
Then, simplify the resulting radical if possible by factoring out perfect squares.
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✔ Problem 1: √2 · √5
→ Multiply inside: √(2·5) = √10
→ 10 has no perfect square factors (other than 1), so it’s simplified.
Answer: √10
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✔ Problem 2: 2√3 · √12
→ First, multiply the radicals: √3 · √12 = √(3·12) = √36
→ √36 = 6
→ Now multiply by the coefficient: 2 · 6 = 12
Answer: 12
*(Alternative approach: Simplify √12 first → √(4·3) = 2√3. Then 2√3 · 2√3 = 4·3 = 12)*
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✔ Problem 3: √3 · √9
→ √9 = 3 (since 3² = 9)
→ So: √3 · 3 = 3√3
Answer: 3√3
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✔ Problem 4: √6 · √6
→ √6 · √6 = (√6)² = 6
Answer: 6
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✔ Problem 5: √18 · √21
→ Multiply inside: √(18·21) = √378
→ Factor 378:
378 = 2 · 189 = 2 · 3 · 63 = 2 · 3 · 3 · 21 = 2 · 3 · 3 · 3 · 7 = 2 · 3³ · 7
→ Pull out pairs: √(3² · 2 · 3 · 7) = 3√(2·3·7) = 3√42
Answer: 3√42
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✔ Problem 6: -3√15 · -4√6
→ Multiply coefficients: (-3) · (-4) = 12
→ Multiply radicals: √15 · √6 = √(15·6) = √90
→ Simplify √90: 90 = 9·10 → √90 = √(9·10) = 3√10
→ Final: 12 · 3√10 = 36√10
Answer: 36√10
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✔ Problem 7: -4√15 · -3√3
→ Coefficients: (-4) · (-3) = 12
→ Radicals: √15 · √3 = √(15·3) = √45
→ Simplify √45: 45 = 9·5 → √45 = 3√5
→ Final: 12 · 3√5 = 36√5
Answer: 36√5
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✔ Problem 8: 3√12 · √6
→ First, simplify √12: √12 = √(4·3) = 2√3
→ So: 3 · 2√3 · √6 = 6 · √3 · √6
→ √3 · √6 = √18
→ √18 = √(9·2) = 3√2
→ Final: 6 · 3√2 = 18√2
Answer: 18√2
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✔ Problem 9: √3 · -√25
→ √25 = 5
→ So: √3 · (-5) = -5√3
Answer: -5√3
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✔ Problem 10: √5 · √10
→ Multiply inside: √(5·10) = √50
→ Simplify √50: 50 = 25·2 → √50 = 5√2
Answer: 5√2
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## 📝 Final Answers:
1.
√10
2.
12
3.
3√3
4.
6
5.
3√42
6.
36√10
7.
36√5
8.
18√2
9.
-5√3
10.
5√2
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Let me know if you’d like a printable version or more practice problems! 😊
Parent Tip: Review the logic above to help your child master the concept of multiply radicals worksheet.
