Multiplying Radicals Worksheets - Math Monks - Free Printable

Let’s solve each problem step by step. We’re multiplying radicals — that means square roots (or sometimes cube roots) being multiplied together.

Remember:
- √a · √b = √(a·b)
- If there are numbers in front, multiply those first.
- Then simplify the radical if possible (look for perfect squares inside).

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Problem 1: √2 · √5

Multiply under one root:
√(2·5) = √10
Can’t simplify further → √10

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Problem 2: 2√3 · √12

First, multiply coefficients and radicals separately:
2 × 1 = 2 (coefficient)
√3 × √12 = √(3×12) = √36

Now: 2 × √36 = 2 × 6 = 12

*(Note: √36 is 6 because 6²=36)*

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Problem 3: ∛3 · ∛9

Cube roots! Same rule: ∛a · ∛b = (a·b)

(3×9) = 27
∛27 = 3 because 3³ = 27 → 3

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Problem 4: √6 · √6

√(6×6) = √36 = 6

Or think: √6 times itself is just 6.

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Problem 5: √18 · √21

√(18×21) = √378

Now simplify √378.

Factor 378:
378 = 2 × 189 = 2 × 3 × 63 = 2 × 3 × 3 × 21 = 2 × 3² × 3 × 7 = 2 × 3³ × 7
Better to group perfect squares:

378 = 9 × 42 → since 9 is a perfect square.

So √378 = √(9×42) = √9 × √42 = 3√42 → 3√42

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Problem 6: -3√15 · -4√6

Multiply coefficients: (-3) × (-4) = +12
Multiply radicals: √15 × √6 = √(15×6) = √90

So we have: 12√90

Simplify √90:
90 = 9 × 10 → √90 = √9 × 10 = 3√10

Then: 12 × 3√10 = 36√10

Wait — no! Wait: 12 × √90 = 12 × 3√10? No!

Actually: 12 × √90 = 12 × (√9 × √10) = 12 × 3 × √10 = 36√10

Yes, correct.

But let me double-check:

Original: -3√15 * -4√6 = (+12) * √(15*6) = 12√90
√90 = √(9*10) = 3√10
12 * 3√10 = 36√10 → ✔

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Problem 7: -4√15 · -3√3

Coefficients: (-4) × (-3) = +12
Radicals: √15 × √3 = √(15×3) = √45

So: 12√45

Simplify √45: 45 = 9×5 → √45 = 3√5

Then: 12 × 3√5 = 36√5

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Problem 8: 3√12 · √6

Coefficient: 3 × 1 = 3
Radical: √12 × √6 = √(12×6) = √72

So: 3√72

Simplify √72: 72 = 36×2 → √72 = 6√2

Then: 3 × 6√2 = 18√2

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Problem 9: √3 · -√25

First, √25 = 5, so this becomes:
√3 × (-5) = -5√3

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Problem 10: √5 · √10

√(5×10) = √50
√50 = √(25×2) = 5√2 → 5√2

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Final Answers:

1. √10
2. 12
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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Final Answer:
1. √10
2. 12
3. 3
4. 6
5. 3√42
6. 36√10
7. 36√5
8. 18√2
9. -5√3
10. 5√2