Multiplying Radical Expressions worksheets - Free Printable

Let's solve each of these multiplying radical expressions step by step.

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Rules to Remember:

1. $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$
2. $\sqrt{a^2} = a$ (if $a \geq 0$)
3. Simplify radicals by factoring out perfect squares.
4. Multiply coefficients separately from radicals.
5. Negative signs are treated like regular multiplication.

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Now, let's go through each problem:

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1) $3\sqrt{12} \cdot \sqrt{6}$

First, simplify $\sqrt{12}$:
$$
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}
$$
So:
$$
3 \cdot 2\sqrt{3} \cdot \sqrt{6} = 6\sqrt{3} \cdot \sqrt{6} = 6\sqrt{18}
$$
Now simplify $\sqrt{18}$:
$$
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
$$
So:
$$
6 \cdot 3\sqrt{2} = 18\sqrt{2}
$$

✔ Answer: $18\sqrt{2}$

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2) $\sqrt{5} \cdot \sqrt{10}$

$$
\sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} = \sqrt{50}
$$
$$
\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
$$

✔ Answer: $5\sqrt{2}$

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3) $\sqrt{6} \cdot \sqrt{6}$

$$
\sqrt{6} \cdot \sqrt{6} = \sqrt{36} = 6
$$

✔ Answer: $6$

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4) $\sqrt{5} \cdot (-4\sqrt{20})$

First, simplify $\sqrt{20}$:
$$
\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
$$
So:
$$
\sqrt{5} \cdot (-4 \cdot 2\sqrt{5}) = \sqrt{5} \cdot (-8\sqrt{5}) = -8 \cdot (\sqrt{5} \cdot \sqrt{5}) = -8 \cdot 5 = -40
$$

✔ Answer: $-40$

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5) $-4\sqrt{15} \cdot (-\sqrt{3})$

Multiply the coefficients and radicals:
$$
(-4) \cdot (-1) = 4
$$
$$
\sqrt{15} \cdot \sqrt{3} = \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}
$$
So:
$$
4 \cdot 3\sqrt{5} = 12\sqrt{5}
$$

✔ Answer: $12\sqrt{5}$

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6) $\sqrt{20x^2} \cdot \sqrt{20x}$

First, simplify each radical:
$$
\sqrt{20x^2} = \sqrt{4 \cdot 5 \cdot x^2} = 2x\sqrt{5}
$$
$$
\sqrt{20x} = \sqrt{4 \cdot 5 \cdot x} = 2\sqrt{5x}
$$
Now multiply:
$$
(2x\sqrt{5}) \cdot (2\sqrt{5x}) = 4x \cdot \sqrt{5} \cdot \sqrt{5x} = 4x \cdot \sqrt{25x} = 4x \cdot 5\sqrt{x} = 20x\sqrt{x}
$$

✔ Answer: $20x\sqrt{x}$

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7) $\sqrt{15n^2} \cdot \sqrt{10n^3}$

Simplify each:
$$
\sqrt{15n^2} = n\sqrt{15}, \quad \sqrt{10n^3} = \sqrt{10 \cdot n^2 \cdot n} = n\sqrt{10n}
$$
Now multiply:
$$
n\sqrt{15} \cdot n\sqrt{10n} = n^2 \cdot \sqrt{15 \cdot 10n} = n^2 \cdot \sqrt{150n}
$$
Simplify $\sqrt{150n}$:
$$
\sqrt{150n} = \sqrt{25 \cdot 6n} = 5\sqrt{6n}
$$
So:
$$
n^2 \cdot 5\sqrt{6n} = 5n^2\sqrt{6n}
$$

✔ Answer: $5n^2\sqrt{6n}$

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8) $-\sqrt{6m} \cdot (-\sqrt{6m^2})$

First, simplify:
$$
-\sqrt{6m} \cdot (-\sqrt{6m^2}) = (+)\sqrt{6m} \cdot \sqrt{6m^2}
$$
Now:
$$
\sqrt{6m} \cdot \sqrt{6m^2} = \sqrt{6m \cdot 6m^2} = \sqrt{36m^3}
$$
$$
\sqrt{36m^3} = \sqrt{36 \cdot m^2 \cdot m} = 6m\sqrt{m}
$$

✔ Answer: $6m\sqrt{m}$

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9) $\sqrt{30} \cdot \sqrt{5}$

$$
\sqrt{30} \cdot \sqrt{5} = \sqrt{150} = \sqrt{25 \cdot 6} = 5\sqrt{6}
$$

✔ Answer: $5\sqrt{6}$

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10) $\sqrt{18} \cdot \sqrt{18}$

$$
\sqrt{18} \cdot \sqrt{18} = (\sqrt{18})^2 = 18
$$

✔ Answer: $18$

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11) $\sqrt{12} \cdot \sqrt{42}$

$$
\sqrt{12} \cdot \sqrt{42} = \sqrt{12 \cdot 42} = \sqrt{504}
$$
Factor 504:
$$
504 = 4 \cdot 126 = 4 \cdot 9 \cdot 14 = 4 \cdot 9 \cdot 2 \cdot 7
$$
So:
$$
\sqrt{504} = \sqrt{4 \cdot 9 \cdot 14} = 2 \cdot 3 \cdot \sqrt{14} = 6\sqrt{14}
$$

✔ Answer: $6\sqrt{14}$

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12) $\sqrt{2} \cdot \sqrt{5}$

$$
\sqrt{2} \cdot \sqrt{5} = \sqrt{10}
$$

✔ Answer: $\sqrt{10}$

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13) $\sqrt{14} \cdot \sqrt{28}$

$$
\sqrt{14} \cdot \sqrt{28} = \sqrt{14 \cdot 28} = \sqrt{392}
$$
Factor 392:
$$
392 = 4 \cdot 98 = 4 \cdot 49 \cdot 2 = 4 \cdot 49 \cdot 2
$$
So:
$$
\sqrt{392} = \sqrt{4 \cdot 49 \cdot 2} = 2 \cdot 7 \cdot \sqrt{2} = 14\sqrt{2}
$$

✔ Answer: $14\sqrt{2}$

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14) $\sqrt{5} \cdot \sqrt{3}$

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

✔ Answer: $\sqrt{15}$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $18\sqrt{2}$ |
| 2) | $5\sqrt{2}$ |
| 3) | $6$ |
| 4) | $-40$ |
| 5) | $12\sqrt{5}$ |
| 6) | $20x\sqrt{x}$ |
| 7) | $5n^2\sqrt{6n}$ |
| 8) | $6m\sqrt{m}$ |
| 9) | $5\sqrt{6}$ |
| 10) | $18$ |
| 11) | $6\sqrt{14}$ |
| 12) | $\sqrt{10}$ |
| 13) | $14\sqrt{2}$ |
| 14) | $\sqrt{15}$ |

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Let me know if you'd like this in a printable format or need explanations for any specific step!