Multiplying Radicals Worksheet, Rules, And Examples - Free Printable

Sure! Let’s solve each of the 8 problems on this “Multiplying Radical Expressions” worksheet step by step.

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Key Rule to Remember:

> √a · √b = √(a·b)
> (The product of two square roots is the square root of the product.)

Also, after multiplying, we simplify the radical if possible by factoring out perfect squares.

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## Problem 1: √3 · √5

Step 1: Multiply under one radical.
→ √(3·5) = √15

Step 2: Check if √15 can be simplified.
15 = 3 × 5 → no perfect square factors other than 1.

✔ Final Answer: √15

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## Problem 2: √7 · √5

Step 1: Multiply under one radical.
→ √(7·5) = √35

Step 2: 35 = 5 × 7 → no perfect square factors.

✔ Final Answer: √35

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## Problem 3: √8 · √2

Step 1: Multiply under one radical.
→ √(8·2) = √16

Step 2: Simplify √16.
→ √16 = 4 (since 4² = 16)

✔ Final Answer: 4

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## Problem 4: √5 · √20

Step 1: Multiply under one radical.
→ √(5·20) = √100

Step 2: Simplify √100.
→ √100 = 10

✔ Final Answer: 10

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## Problem 5: √2 · √20

Step 1: Multiply under one radical.
→ √(2·20) = √40

Step 2: Factor 40 to find perfect squares.
→ 40 = 4 × 10 → √40 = √(4·10) = √4 · √10 = 2√10

✔ Final Answer: 2√10

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## Problem 6: √12 · √9

Step 1: Multiply under one radical.
→ √(12·9) = √108

Step 2: Factor 108 to find perfect squares.
→ 108 = 36 × 3 → √108 = √(36·3) = √36 · √3 = 6√3

✔ Final Answer: 6√3

*(Alternative approach: Since √9 = 3, you could do √12 · 3 = 3√12 = 3·√(4·3) = 3·2√3 = 6√3 — same result.)*

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## Problem 7: √20 · √30

Step 1: Multiply under one radical.
→ √(20·30) = √600

Step 2: Factor 600 to find perfect squares.
→ 600 = 100 × 6 → √600 = √(100·6) = √100 · √6 = 10√6

✔ Final Answer: 10√6

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## Problem 8: √18 · √24

Step 1: Multiply under one radical.
→ √(18·24) = √432

Step 2: Factor 432 to find perfect squares.
Let’s factor it step by step:

432 ÷ 2 = 216
216 ÷ 2 = 108
108 ÷ 2 = 54
54 ÷ 2 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1

So, 432 = 2⁴ × 3³
= (2²)² × 3² × 3
= 16 × 9 × 3 → so √432 = √(16·9·3) = √16 · √9 · √3 = 4·3·√3 = 12√3

✔ Final Answer: 12√3

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## ✔ All Answers Summary:

1) √15
2) √35
3) 4
4) 10
5) 2√10
6) 6√3
7) 10√6
8) 12√3

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