Simplifying Radicals (B) Worksheet | Fun and Engaging Algebra I ... - Free Printable

Let’s solve each problem step by step. We’ll use the rules of radicals:

- √a × b = √(a×b)
- √a ÷ b = √(a÷b)
- √a × √a = a (because square root and squaring cancel out)
- Simplify radicals by factoring out perfect squares when needed.

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Section A: Multiply Radicals

1) √6 × √2 = √(6×2) = √12 → but we can simplify √12 = √(4×3) = 2√3
Wait — actually, let’s check if we should leave as √12 or simplify. The instruction says “simplify”, so yes, simplify fully.

But wait — maybe some problems are designed to give whole numbers? Let’s recalculate carefully.

Actually, for multiplication, sometimes it’s easier to multiply first then simplify.

Let me go one by one with care.

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Section A

1) √6 × 2 = √(6×2) = √12 = √(4×3) = 2√3

But hold on — looking at other problems like #3: √7 × √7 = 7. So maybe they expect simplified radical form or integer if possible.

Let’s do all Section A:

1) √6 × 2 = √12 = 2√3
→ But let me double-check: is there a better way? No, that’s correct.

Wait — perhaps I made a mistake in approach. Let’s look at problem 5: √3 × √3 = 3. That’s straightforward.

Problem 9: √4 × √4 × √4 = 2 × 2 × 2 = 8

So maybe for some, we compute numerically.

Let me redo Section A with this mindset: simplify completely, whether to integer or simplified radical.

1) √6 × √2 = √12 = 2√3 ✔

2) √20 × √5 = (20×5) = √100 = 10

3) √7 × √7 = 7

4) √2 × √72 = √(2×72) = √144 = 12

5) √3 × √3 = 3

6) √4 × √16 = 2 × 4 = 8

7) √4 × 5 × √5 = 2 × (√5 × √5) = 2 × 5 = 10

8) √2 × √4 × √2 = (√2 × √2) × √4 = 2 × 2 = 4

9) √4 × √4 × √4 = 2 × 2 × 2 = 8

10) √3 × √6 × √3 = (√3 × √3) × √6 = 3 × √6 = 3√6

Wait — but let’s check: √3 × √6 = √18, then × √3 = √54? No, better to group same radicals.

Actually: √3 × √3 × √6 = 3 × √6 → yes, 3√6

But is that simplified? Yes.

Alternatively, √3 × 6 × √3 = (3×6×3) = √54 = √(9×6) = 3√6 → same.

11) √6 × √12 × √6 = (√6 × √6) × √12 = 6 × √12 = 6 × 2√3 = 12√3

Or: √(6×12×6) = √432 = √(144×3) = 12√3 → same.

12) √2 × √12 × √3 = (2×12×3) = √72 = √(36×2) = 6√2

Okay, now Section B: Division and mixed operations.

Rules:
- √a ÷ √b = (a/b)
- Also, √a × √b ÷ √c = √(a×b/c)

Section B:

1) √3 ÷ √3 = √(3/3) = √1 = 1

2) √15 ÷ √5 = (15/5) = √3 → √3

3) √8 ÷ √2 = √(8/2) = √4 = 2

4) √27 ÷ √3 = √(27/3) = √9 = 3

5) √48 √3 = √(48/3) = √16 = 4

6) √54 ÷ √9 = (54/9) = √6 → √6

Wait — √9 is 3, so √54 ÷ 3 = (3√6)/3 = √6 → same.

7) √96 √12 = √(96/12) = √8 = √(4×2) = 2√2

8) √72 √18 = √(72/18) = √4 = 2

9) √48 ÷ √8 = √(48/8) = √6 → √6

10) √5 × √6 ÷ √2 = (5×6 / 2) = √(30/2) = √15 → √15

11) √32 × √10 √5 = √(32×10 / 5) = √(320/5) = √64 = 8

Check: 32×10=320, 320÷5=64, √64=8 → yes.

12) √24 × √2 ÷ √3 = √(24×2 / 3) = √(48/3) = √16 = 4

Perfect.

Now let’s compile final answers clearly.

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Final Answer:

Section A

1) 2√3
2) 10
3) 7
4) 12
5) 3
6) 8
7) 10
8) 4
9) 8
10) 3√6
11) 12√3
12) 6√2

Section B

1) 1
2) √3
3) 2
4) 3
5) 4
6) √6
7) 2√2
8) 2
9) √6
10) √15
11) 8
12) 4