Simplifying Radicals worksheet with multiplication and division exercises for algebra practice.

Let's solve each problem step by step, using the properties of radicals:

Key Rules:

1. $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$
2. $\sqrt{a} \div \sqrt{b} = \sqrt{a \div b}$
3. $\sqrt{a^2} = a$ (for $a \geq 0$)
4. Simplify square roots by factoring perfect squares.

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## ✔ Section A: Multiplication of Radicals

We’ll simplify each expression using multiplication rules.

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1) $\sqrt{6} \times \sqrt{2}$

$$
\sqrt{6 \times 2} = \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
$$

✔ Answer: $2\sqrt{3}$

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2) $\sqrt{20} \times \sqrt{5}$

$$
\sqrt{20 \times 5} = \sqrt{100} = 10
$$

✔ Answer: $10$

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3) $\sqrt{7} \times \sqrt{7}$

$$
\sqrt{7 \times 7} = \sqrt{49} = 7
$$

✔ Answer: $7$

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4) $\sqrt{2} \times \sqrt{72}$

$$
\sqrt{2 \times 72} = \sqrt{144} = 12
$$

✔ Answer: $12$

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5) $\sqrt{3} \times \sqrt{3}$

$$
\sqrt{3 \times 3} = \sqrt{9} = 3
$$

✔ Answer: $3$

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6) $\sqrt{4} \times \sqrt{16}$

$$
\sqrt{4} = 2,\quad \sqrt{16} = 4,\quad 2 \times 4 = 8
$$

Or: $\sqrt{4 \times 16} = \sqrt{64} = 8$

✔ Answer: $8$

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7) $\sqrt{4} \times \sqrt{5} \times \sqrt{5}$

$$
\sqrt{4} = 2,\quad \sqrt{5} \times \sqrt{5} = 5,\quad 2 \times 5 = 10
$$

✔ Answer: $10$

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8) $\sqrt{2} \times \sqrt{4} \times \sqrt{2}$

$$
\sqrt{2} \times \sqrt{2} = 2,\quad \sqrt{4} = 2,\quad 2 \times 2 = 4
$$

✔ Answer: $4$

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9) $\sqrt{4} \times \sqrt{4} \times \sqrt{4}$

$$
\sqrt{4} = 2,\quad 2 \times 2 \times 2 = 8
$$

✔ Answer: $8$

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10) $\sqrt{3} \times \sqrt{6} \times \sqrt{3}$

Group: $\sqrt{3} \times \sqrt{3} = 3$, then $3 \times \sqrt{6} = 3\sqrt{6}$

Or: $\sqrt{3 \times 6 \times 3} = \sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$

✔ Answer: $3\sqrt{6}$

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11) $\sqrt{6} \times \sqrt{12} \times \sqrt{6}$

First: $\sqrt{6} \times \sqrt{6} = 6$, then $6 \times \sqrt{12}$

Now: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$, so $6 \times 2\sqrt{3} = 12\sqrt{3}$

Alternatively: $\sqrt{6 \times 12 \times 6} = \sqrt{432} = \sqrt{144 \times 3} = 12\sqrt{3}$

✔ Answer: $12\sqrt{3}$

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12) $\sqrt{2} \times \sqrt{12} \times \sqrt{3}$

$\sqrt{12} = 2\sqrt{3}$, so:

$$
\sqrt{2} \times 2\sqrt{3} \times \sqrt{3} = 2 \times \sqrt{2} \times \sqrt{3} \times \sqrt{3} = 2 \times \sqrt{2} \times 3 = 6\sqrt{2}
$$

Or: $\sqrt{2 \times 12 \times 3} = \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

✔ Answer: $6\sqrt{2}$

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## ✔ Section B: Division and Mixed Operations

Use: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

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1) $\sqrt{3} \div \sqrt{3}$

$$
= \sqrt{\frac{3}{3}} = \sqrt{1} = 1
$$

✔ Answer: $1$

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2) $\sqrt{15} \div \sqrt{5}$

$$
= \sqrt{\frac{15}{5}} = \sqrt{3}
$$

✔ Answer: $\sqrt{3}$

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3) $\sqrt{8} \div \sqrt{2}$

$$
= \sqrt{\frac{8}{2}} = \sqrt{4} = 2
$$

✔ Answer: $2$

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4) $\sqrt{27} \div \sqrt{3}$

$$
= \sqrt{\frac{27}{3}} = \sqrt{9} = 3
$$

✔ Answer: $3$

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5) $\sqrt{48} \div \sqrt{3}$

$$
= \sqrt{\frac{48}{3}} = \sqrt{16} = 4
$$

✔ Answer: $4$

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6) $\sqrt{54} \div \sqrt{9}$

$$
= \sqrt{\frac{54}{9}} = \sqrt{6}
$$

✔ Answer: $\sqrt{6}$

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7) $\sqrt{96} \div \sqrt{12}$

$$
= \sqrt{\frac{96}{12}} = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
$$

✔ Answer: $2\sqrt{2}$

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8) $\sqrt{72} \div \sqrt{18}$

$$
= \sqrt{\frac{72}{18}} = \sqrt{4} = 2
$$

✔ Answer: $2$

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9) $\sqrt{48} \div \sqrt{8}$

$$
= \sqrt{\frac{48}{8}} = \sqrt{6}
$$

✔ Answer: $\sqrt{6}$

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10) $\sqrt{5} \times \sqrt{6} \div \sqrt{2}$

First: $\sqrt{5} \times \sqrt{6} = \sqrt{30}$, then divide by $\sqrt{2}$:

$$
\frac{\sqrt{30}}{\sqrt{2}} = \sqrt{\frac{30}{2}} = \sqrt{15}
$$

✔ Answer: $\sqrt{15}$

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11) $\sqrt{32} \times \sqrt{10} \div \sqrt{5}$

First: $\sqrt{32} \times \sqrt{10} = \sqrt{320}$, then divide by $\sqrt{5}$:

$$
\frac{\sqrt{320}}{\sqrt{5}} = \sqrt{\frac{320}{5}} = \sqrt{64} = 8
$$

✔ Answer: $8$

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12) $\sqrt{24} \times \sqrt{2} \div \sqrt{3}$

First: $\sqrt{24} \times \sqrt{2} = \sqrt{48}$, then divide by $\sqrt{3}$:

$$
\frac{\sqrt{48}}{\sqrt{3}} = \sqrt{\frac{48}{3}} = \sqrt{16} = 4
$$

✔ Answer: $4$

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## ✔ Final Answers Summary:

Section A:

1. $2\sqrt{3}$
2. $10$
3. $7$
4. $12$
5. $3$
6. $8$
7. $10$
8. $4$
9. $8$
10. $3\sqrt{6}$
11. $12\sqrt{3}$
12. $6\sqrt{2}$

Section B:

1. $1$
2. $\sqrt{3}$
3. $2$
4. $3$
5. $4$
6. $\sqrt{6}$
7. $2\sqrt{2}$
8. $2$
9. $\sqrt{6}$
10. $\sqrt{15}$
11. $8$
12. $4$

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