Simplifying Radical Expressions worksheets - Free Printable

Let's solve each of these radical expressions step by step. The goal is to simplify each square root expression by factoring out perfect squares.

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Key Concept:

To simplify $\sqrt{a \cdot b}$, we use:
$$
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
$$
We look for perfect square factors (like 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.) and pull them out of the square root.

Also, recall:
- $\sqrt{x^2} = x$ (if $x \geq 0$)
- $\sqrt{x^3} = \sqrt{x^2 \cdot x} = x\sqrt{x}$

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Now let’s go through each problem:

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1) $\sqrt{125n}$

Factor 125:
$125 = 25 \times 5 = 5^2 \times 5$

So:
$$
\sqrt{125n} = \sqrt{25 \cdot 5 \cdot n} = \sqrt{25} \cdot \sqrt{5n} = 5\sqrt{5n}
$$

✔ Answer: $5\sqrt{5n}$

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2) $\sqrt{216v}$

Factor 216:
$216 = 36 \times 6 = 6^2 \times 6$, but better:
$216 = 36 \cdot 6 = (6^2) \cdot 6$, so:
$$
\sqrt{216v} = \sqrt{36 \cdot 6 \cdot v} = \sqrt{36} \cdot \sqrt{6v} = 6\sqrt{6v}
$$

✔ Answer: $6\sqrt{6v}$

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3) $\sqrt{512k^2}$

Factor 512:
$512 = 256 \times 2 = 16^2 \times 2 = (16)^2 \cdot 2$, or better:
$512 = 64 \times 8 = 64 \cdot 8$, but 64 is a perfect square.

Actually: $512 = 64 \cdot 8$, and $64 = 8^2$, so:
$$
\sqrt{512k^2} = \sqrt{64 \cdot 8 \cdot k^2} = \sqrt{64} \cdot \sqrt{k^2} \cdot \sqrt{8} = 8 \cdot k \cdot \sqrt{8}
$$
Now simplify $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$

So:
$$
8k \cdot 2\sqrt{2} = 16k\sqrt{2}
$$

✔ Answer: $16k\sqrt{2}$

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4) $\sqrt{512m^3}$

Same as above, $512 = 64 \cdot 8$, and $m^3 = m^2 \cdot m$

So:
$$
\sqrt{512m^3} = \sqrt{64 \cdot 8 \cdot m^2 \cdot m} = \sqrt{64} \cdot \sqrt{m^2} \cdot \sqrt{8m} = 8 \cdot m \cdot \sqrt{8m}
$$
Now $\sqrt{8m} = \sqrt{4 \cdot 2m} = 2\sqrt{2m}$

So:
$$
8m \cdot 2\sqrt{2m} = 16m\sqrt{2m}
$$

✔ Answer: $16m\sqrt{2m}$

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5) $\sqrt{216k^4}$

First, $216 = 36 \cdot 6$, and $k^4 = (k^2)^2$

So:
$$
\sqrt{216k^4} = \sqrt{36 \cdot 6 \cdot k^4} = \sqrt{36} \cdot \sqrt{k^4} \cdot \sqrt{6} = 6 \cdot k^2 \cdot \sqrt{6} = 6k^2\sqrt{6}
$$

✔ Answer: $6k^2\sqrt{6}$

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6) $\sqrt{100p^3}$

$100 = 10^2$, $p^3 = p^2 \cdot p$

So:
$$
\sqrt{100p^3} = \sqrt{100} \cdot \sqrt{p^2} \cdot \sqrt{p} = 10 \cdot p \cdot \sqrt{p} = 10p\sqrt{p}
$$

✔ Answer: $10p\sqrt{p}$

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7) $\sqrt{80p^3}$

Factor 80: $80 = 16 \cdot 5$, and $p^3 = p^2 \cdot p$

So:
$$
\sqrt{80p^3} = \sqrt{16 \cdot 5 \cdot p^2 \cdot p} = \sqrt{16} \cdot \sqrt{p^2} \cdot \sqrt{5p} = 4 \cdot p \cdot \sqrt{5p} = 4p\sqrt{5p}
$$

✔ Answer: $4p\sqrt{5p}$

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8) $\sqrt{45p^2}$

$45 = 9 \cdot 5$, $p^2$ is perfect square

$$
\sqrt{45p^2} = \sqrt{9 \cdot 5 \cdot p^2} = \sqrt{9} \cdot \sqrt{p^2} \cdot \sqrt{5} = 3 \cdot p \cdot \sqrt{5} = 3p\sqrt{5}
$$

✔ Answer: $3p\sqrt{5}$

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9) $\sqrt{147m^3}$

$147 = 49 \cdot 3$, $m^3 = m^2 \cdot m$

$$
\sqrt{147m^3} = \sqrt{49 \cdot 3 \cdot m^2 \cdot m} = \sqrt{49} \cdot \sqrt{m^2} \cdot \sqrt{3m} = 7 \cdot m \cdot \sqrt{3m} = 7m\sqrt{3m}
$$

✔ Answer: $7m\sqrt{3m}$

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10) $\sqrt{100m^4}$

$100 = 10^2$, $m^4 = (m^2)^2$

$$
\sqrt{100m^4} = \sqrt{100} \cdot \sqrt{m^4} = 10 \cdot m^2 = 10m^2
$$

✔ Answer: $10m^2$

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11) $\sqrt{75x^2}$

$75 = 25 \cdot 3$, $x^2$ is perfect square

$$
\sqrt{75x^2} = \sqrt{25 \cdot 3 \cdot x^2} = \sqrt{25} \cdot \sqrt{x^2} \cdot \sqrt{3} = 5 \cdot x \cdot \sqrt{3} = 5x\sqrt{3}
$$

✔ Answer: $5x\sqrt{3}$

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12) $\sqrt{64m^4}$

$64 = 8^2$, $m^4 = (m^2)^2$

$$
\sqrt{64m^4} = \sqrt{64} \cdot \sqrt{m^4} = 8 \cdot m^2 = 8m^2
$$

✔ Answer: $8m^2$

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13) $\sqrt{16u^4}$

$16 = 4^2$, $u^4 = (u^2)^2$

$$
\sqrt{16u^4} = \sqrt{16} \cdot \sqrt{u^4} = 4 \cdot u^2 = 4u^2
$$

✔ Answer: $4u^2$

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14) $\sqrt{28x^3}$

$28 = 4 \cdot 7$, $x^3 = x^2 \cdot x$

$$
\sqrt{28x^3} = \sqrt{4 \cdot 7 \cdot x^2 \cdot x} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{7x} = 2 \cdot x \cdot \sqrt{7x} = 2x\sqrt{7x}
$$

✔ Answer: $2x\sqrt{7x}$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) $\sqrt{125n}$ | $5\sqrt{5n}$ |
| 2) $\sqrt{216v}$ | $6\sqrt{6v}$ |
| 3) $\sqrt{512k^2}$ | $16k\sqrt{2}$ |
| 4) $\sqrt{512m^3}$ | $16m\sqrt{2m}$ |
| 5) $\sqrt{216k^4}$ | $6k^2\sqrt{6}$ |
| 6) $\sqrt{100p^3}$ | $10p\sqrt{p}$ |
| 7) $\sqrt{80p^3}$ | $4p\sqrt{5p}$ |
| 8) $\sqrt{45p^2}$ | $3p\sqrt{5}$ |
| 9) $\sqrt{147m^3}$ | $7m\sqrt{3m}$ |
| 10) $\sqrt{100m^4}$ | $10m^2$ |
| 11) $\sqrt{75x^2}$ | $5x\sqrt{3}$ |
| 12) $\sqrt{64m^4}$ | $8m^2$ |
| 13) $\sqrt{16u^4}$ | $4u^2$ |
| 14) $\sqrt{28x^3}$ | $2x\sqrt{7x}$ |

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