Sure! Let’s solve each problem step by step. The key rule for multiplying radicals is:
>
√a · √b = √(a·b) — *as long as a and b are non-negative.*
Also, if there are coefficients (numbers in front), multiply them separately.
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Problem 1: √2 · √5
Use the rule:
√2 · √5 = √(2·5) =
√10
✔ Answer: √10
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Problem 2: 2√3 · √12
First, multiply coefficients: 2 · 1 = 2
Then multiply radicals: √3 · √12 = √(3·12) = √36 = 6
Now multiply: 2 · 6 =
12
✔ Answer: 12
*(Alternative way: Simplify √12 first → √(4·3) = 2√3, so 2√3 · 2√3 = 4 · (√3)² = 4·3 = 12)*
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Problem 3: √3 · √9
√9 = 3, so:
√3 · 3 =
3√3
Or using the rule: √3 · √9 = √(3·9) = √27 = √(9·3) = 3√3
✔ Answer: 3√3
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Problem 4: √6 · √6
√6 · √6 = √(6·6) = √36 =
6
✔ Answer: 6
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Problem 5: √18 · √21
√18 · √21 = √(18·21)
Calculate 18·21 = 378
Now simplify √378:
Factor 378:
378 = 2 × 189 = 2 × 3 × 63 = 2 × 3 × 3 × 21 = 2 × 3 × 3 × 3 × 7 = 2 × 3³ × 7
So √378 = √(9 × 42) = √9 × √42 =
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 = 3√10
So total: 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: √(9×5) =
3√5
Total: 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 = 6 · √(3×6) = 6 · √18
Simplify √18: √(9×2) = 3√2
Total: 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
√5 · √10 = √(5×10) = √50
Simplify √50: √(25×2) =
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
Let me know if you’d like to see any step explained further!
Parent Tip: Review the logic above to help your child master the concept of multiplying radicals worksheet.
