Worksheet for simplifying the multiplication of radicals, including various square and cube roots.

Sure! Let’s solve each problem step by step. The key rule for multiplying radicals is:

> √a · √b = √(a·b), provided a and b are non-negative.

Also, if there are coefficients (numbers in front of the radicals), 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

*(Alternatively: √12 = 2√3, so 2√3 · 2√3 = 4·3 = 12)*

✔ Answer: 12

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Problem 3: √3 · √9

√9 = 3, so:
√3 · 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

Multiply under one radical:
√(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 = √9 × √10 = 3√10

Now multiply: 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

Multiply: 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 expression becomes: 3 × 2√3 × √6 = 6√3 · √6

Now: √3 · √6 = √18 = √(9×2) = 3√2

So: 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:
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

Let me know if you’d like to see any step explained further!