Simplifying Radicals worksheet with 18 algebraic problems for students to solve.

Let's solve each of these radical simplification problems step by step. The goal is to simplify each expression by factoring out perfect squares from under the square root and then simplifying.

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1) $-2\sqrt{192}$

Factor 192:
$$
192 = 64 \times 3 = (8^2) \times 3
$$
$$
\sqrt{192} = \sqrt{64 \times 3} = \sqrt{64} \cdot \sqrt{3} = 8\sqrt{3}
$$
Now multiply by -2:
$$
-2 \cdot 8\sqrt{3} = -16\sqrt{3}
$$

✔ Answer: $-16\sqrt{3}$

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2) $-5\sqrt{108}$

Factor 108:
$$
108 = 36 \times 3 = (6^2) \times 3
$$
$$
\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}
$$
Multiply by -5:
$$
-5 \cdot 6\sqrt{3} = -30\sqrt{3}
$$

✔ Answer: $-30\sqrt{3}$

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3) $-2\sqrt{50}$

Factor 50:
$$
50 = 25 \times 2 = (5^2) \times 2
$$
$$
\sqrt{50} = 5\sqrt{2}
$$
Multiply by -2:
$$
-2 \cdot 5\sqrt{2} = -10\sqrt{2}
$$

✔ Answer: $-10\sqrt{2}$

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4) $8\sqrt{128}$

Factor 128:
$$
128 = 64 \times 2 = (8^2) \times 2
$$
$$
\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}
$$
Multiply by 8:
$$
8 \cdot 8\sqrt{2} = 64\sqrt{2}
$$

✔ Answer: $64\sqrt{2}$

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5) $-4\sqrt{72}$

Factor 72:
$$
72 = 36 \times 2 = (6^2) \times 2
$$
$$
\sqrt{72} = 6\sqrt{2}
$$
Multiply by -4:
$$
-4 \cdot 6\sqrt{2} = -24\sqrt{2}
$$

✔ Answer: $-24\sqrt{2}$

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6) $2\sqrt{98}$

Factor 98:
$$
98 = 49 \times 2 = (7^2) \times 2
$$
$$
\sqrt{98} = 7\sqrt{2}
$$
Multiply by 2:
$$
2 \cdot 7\sqrt{2} = 14\sqrt{2}
$$

✔ Answer: $14\sqrt{2}$

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7) $4\sqrt{72}$

We already did $\sqrt{72} = 6\sqrt{2}$

$$
4 \cdot 6\sqrt{2} = 24\sqrt{2}
$$

✔ Answer: $24\sqrt{2}$

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8) $-4\sqrt{252}$

Factor 252:
$$
252 = 36 \times 7 = (6^2) \times 7
$$
$$
\sqrt{252} = 6\sqrt{7}
$$
Multiply by -4:
$$
-4 \cdot 6\sqrt{7} = -24\sqrt{7}
$$

✔ Answer: $-24\sqrt{7}$

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9) $-2\sqrt{28}$

Factor 28:
$$
28 = 4 \times 7 = (2^2) \times 7
$$
$$
\sqrt{28} = 2\sqrt{7}
$$
Multiply by -2:
$$
-2 \cdot 2\sqrt{7} = -4\sqrt{7}
$$

✔ Answer: $-4\sqrt{7}$

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10) $-\sqrt{216}$

Factor 216:
$$
216 = 36 \times 6 = (6^2) \times 6
$$
$$
\sqrt{216} = \sqrt{36 \times 6} = 6\sqrt{6}
$$
Apply negative sign:
$$
-6\sqrt{6}
$$

✔ Answer: $-6\sqrt{6}$

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11) $-6\sqrt{384}$

Factor 384:
$$
384 = 64 \times 6 = (8^2) \times 6
$$
$$
\sqrt{384} = 8\sqrt{6}
$$
Multiply by -6:
$$
-6 \cdot 8\sqrt{6} = -48\sqrt{6}
$$

✔ Answer: $-48\sqrt{6}$

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12) $2\sqrt{96}$

Factor 96:
$$
96 = 16 \times 6 = (4^2) \times 6
$$
$$
\sqrt{96} = 4\sqrt{6}
$$
Multiply by 2:
$$
2 \cdot 4\sqrt{6} = 8\sqrt{6}
$$

✔ Answer: $8\sqrt{6}$

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13) $4\sqrt{384}$

We already found $\sqrt{384} = 8\sqrt{6}$

$$
4 \cdot 8\sqrt{6} = 32\sqrt{6}
$$

✔ Answer: $32\sqrt{6}$

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14) $5\sqrt{80}$

Factor 80:
$$
80 = 16 \times 5 = (4^2) \times 5
$$
$$
\sqrt{80} = 4\sqrt{5}
$$
Multiply by 5:
$$
5 \cdot 4\sqrt{5} = 20\sqrt{5}
$$

✔ Answer: $20\sqrt{5}$

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15) $7\sqrt{245}$

Factor 245:
$$
245 = 49 \times 5 = (7^2) \times 5
$$
$$
\sqrt{245} = 7\sqrt{5}
$$
Multiply by 7:
$$
7 \cdot 7\sqrt{5} = 49\sqrt{5}
$$

✔ Answer: $49\sqrt{5}$

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16) $8\sqrt{36}$

$\sqrt{36} = 6$

$$
8 \cdot 6 = 48
$$

✔ Answer: $48$ (no radical)

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17) $-8\sqrt{144}$

$\sqrt{144} = 12$

$$
-8 \cdot 12 = -96
$$

✔ Answer: $-96$

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18) $2\sqrt{16}$

$\sqrt{16} = 4$

$$
2 \cdot 4 = 8
$$

✔ Answer: $8$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $-16\sqrt{3}$ |
| 2) | $-30\sqrt{3}$ |
| 3) | $-10\sqrt{2}$ |
| 4) | $64\sqrt{2}$ |
| 5) | $-24\sqrt{2}$ |
| 6) | $14\sqrt{2}$ |
| 7) | $24\sqrt{2}$ |
| 8) | $-24\sqrt{7}$ |
| 9) | $-4\sqrt{7}$ |
| 10) | $-6\sqrt{6}$ |
| 11) | $-48\sqrt{6}$ |
| 12) | $8\sqrt{6}$ |
| 13) | $32\sqrt{6}$ |
| 14) | $20\sqrt{5}$ |
| 15) | $49\sqrt{5}$ |
| 16) | $48$ |
| 17) | $-96$ |
| 18) | $8$ |

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