Practice your algebra skills with this worksheet featuring 14 problems on simplifying radical expressions involving variables.

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.

We'll use the rule:
$$
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
$$
and
$$
\sqrt{x^n} = x^{n/2} \quad \text{(for even powers)}
$$

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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 $
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 $, but better:
$ 512 = 64 \times 8 = 8^2 \times 8 $? Let's do it properly.

Actually:
$ 512 = 2^9 $ → so $ \sqrt{512} = \sqrt{2^9} = \sqrt{2^8 \cdot 2} = 2^4 \sqrt{2} = 16\sqrt{2} $

But we have $ k^2 $, so:
$$
\sqrt{512k^2} = \sqrt{512} \cdot \sqrt{k^2} = 16\sqrt{2} \cdot k = 16k\sqrt{2}
$$

✔ Answer: $ 16k\sqrt{2} $

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

Again, $ 512 = 2^9 $, so $ \sqrt{512} = 16\sqrt{2} $

Now $ m^3 = m^2 \cdot m $, so:
$$
\sqrt{512m^3} = \sqrt{512} \cdot \sqrt{m^3} = 16\sqrt{2} \cdot \sqrt{m^2 \cdot m} = 16\sqrt{2} \cdot m\sqrt{m} = 16m\sqrt{2m}
$$

✔ Answer: $ 16m\sqrt{2m} $

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

$ 216 = 36 \times 6 = 6^2 \times 6 $, so $ \sqrt{216} = 6\sqrt{6} $

$ k^4 = (k^2)^2 $, so $ \sqrt{k^4} = k^2 $

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

✔ Answer: $ 6k^2\sqrt{6} $

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

$ \sqrt{100} = 10 $, $ v^3 = v^2 \cdot v $, so $ \sqrt{v^3} = v\sqrt{v} $

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

✔ Answer: $ 10v\sqrt{v} $

---

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

Factor 80: $ 80 = 16 \times 5 $, so $ \sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5} $

$ p^3 = p^2 \cdot p $, so $ \sqrt{p^3} = p\sqrt{p} $

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

✔ Answer: $ 4p\sqrt{5p} $

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

$ 45 = 9 \times 5 $, so $ \sqrt{45} = 3\sqrt{5} $

$ \sqrt{p^2} = p $

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

✔ Answer: $ 3p\sqrt{5} $

---

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

$ 147 = 49 \times 3 = 7^2 \times 3 $, so $ \sqrt{147} = 7\sqrt{3} $

$ m^3 = m^2 \cdot m $, so $ \sqrt{m^3} = m\sqrt{m} $

So:
$$
\sqrt{147m^3} = 7\sqrt{3} \cdot m\sqrt{m} = 7m\sqrt{3m}
$$

✔ Answer: $ 7m\sqrt{3m} $

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

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

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

✔ Answer: $ 10m^2 $

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

$ 75 = 25 \times 3 $, so $ \sqrt{75} = 5\sqrt{3} $

$ \sqrt{x^2} = x $

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

✔ Answer: $ 5x\sqrt{3} $

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

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

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

✔ Answer: $ 8m^2 $

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

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

So:
$$
\sqrt{16u^4} = 4u^2
$$

✔ Answer: $ 4u^2 $

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

$ 28 = 4 \times 7 $, so $ \sqrt{28} = 2\sqrt{7} $

$ x^3 = x^2 \cdot x $, so $ \sqrt{x^3} = x\sqrt{x} $

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

✔ Answer: $ 2x\sqrt{7x} $

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✔ Final Answers:

1) $ 5\sqrt{5n} $
2) $ 6\sqrt{6v} $
3) $ 16k\sqrt{2} $
4) $ 16m\sqrt{2m} $
5) $ 6k^2\sqrt{6} $
6) $ 10v\sqrt{v} $
7) $ 4p\sqrt{5p} $
8) $ 3p\sqrt{5} $
9) $ 7m\sqrt{3m} $
10) $ 10m^2 $
11) $ 5x\sqrt{3} $
12) $ 8m^2 $
13) $ 4u^2 $
14) $ 2x\sqrt{7x} $

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