Simplifying Radical Expressions worksheet with 14 algebra problems involving variables and exponents.

Of course! I'll solve each of the 14 problems on the worksheet, simplifying the radical expressions step by step. The goal is to factor out perfect squares from under the square root and simplify.

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1) $\sqrt{125n}$

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

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

✔ Answer: $5\sqrt{5n}$

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2) $\sqrt{216v}$

Factor 216: $216 = 36 \cdot 6 = 6^2 \cdot 6$

$\sqrt{216v} = \sqrt{36 \cdot 6 \cdot v} = \sqrt{36} \cdot \sqrt{6v} = 6\sqrt{6v}$

✔ Answer: $6\sqrt{6v}$

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

Factor 512: $512 = 256 \cdot 2 = 16^2 \cdot 2$
Also, $\sqrt{k^2} = |k|$, but since this is algebra 1 and no domain restriction is given, we assume $k \geq 0$, so $\sqrt{k^2} = k$

$\sqrt{512k^2} = \sqrt{256 \cdot 2 \cdot k^2} = \sqrt{256} \cdot \sqrt{2} \cdot \sqrt{k^2} = 16k\sqrt{2}$

✔ Answer: $16k\sqrt{2}$

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

Factor 512: $512 = 256 \cdot 2 = 16^2 \cdot 2$
$m^3 = m^2 \cdot m$

$\sqrt{512m^3} = \sqrt{256 \cdot 2 \cdot m^2 \cdot m} = \sqrt{256} \cdot \sqrt{m^2} \cdot \sqrt{2m} = 16m\sqrt{2m}$

✔ Answer: $16m\sqrt{2m}$

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5) $\sqrt{216k^8}$

Factor 216: $216 = 36 \cdot 6 = 6^2 \cdot 6$
$k^8 = (k^4)^2$

$\sqrt{216k^8} = \sqrt{36 \cdot 6 \cdot (k^4)^2} = \sqrt{36} \cdot \sqrt{(k^4)^2} \cdot \sqrt{6} = 6k^4\sqrt{6}$

✔ Answer: $6k^4\sqrt{6}$

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

$100 = 10^2$
$v^3 = v^2 \cdot v$

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

✔ Answer: $10v\sqrt{v}$

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

Factor 80: $80 = 16 \cdot 5 = 4^2 \cdot 5$
$p^3 = p^2 \cdot p$

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

✔ Answer: $4p\sqrt{5p}$

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8) $\sqrt{45p^7}$

Factor 45: $45 = 9 \cdot 5 = 3^2 \cdot 5$
$p^7 = p^6 \cdot p = (p^3)^2 \cdot p$

$\sqrt{45p^7} = \sqrt{9 \cdot 5 \cdot (p^3)^2 \cdot p} = \sqrt{9} \cdot \sqrt{(p^3)^2} \cdot \sqrt{5p} = 3p^3\sqrt{5p}$

✔ Answer: $3p^3\sqrt{5p}$

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9) $\sqrt{147m^2n^3}$

Factor 147: $147 = 49 \cdot 3 = 7^2 \cdot 3$
$n^3 = n^2 \cdot n$

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

✔ Answer: $7mn\sqrt{3n}$

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10) $\sqrt{200m^4n}$

Factor 200: $200 = 100 \cdot 2 = 10^2 \cdot 2$
$m^4 = (m^2)^2$

$\sqrt{200m^4n} = \sqrt{100 \cdot 2 \cdot (m^2)^2 \cdot n} = \sqrt{100} \cdot \sqrt{(m^2)^2} \cdot \sqrt{2n} = 10m^2\sqrt{2n}$

✔ Answer: $10m^2\sqrt{2n}$

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

Factor 75: $75 = 25 \cdot 3 = 5^2 \cdot 3$
$\sqrt{x^2} = x$ (assuming $x \geq 0$)

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

✔ Answer: $5x\sqrt{3y}$

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12) $\sqrt{64m^2n^3}$

$64 = 8^2$
$n^3 = n^2 \cdot n$

$\sqrt{64m^2n^3} = \sqrt{64 \cdot m^2 \cdot n^2 \cdot n} = \sqrt{64} \cdot \sqrt{m^2} \cdot \sqrt{n^2} \cdot \sqrt{n} = 8mn\sqrt{n}$

✔ Answer: $8mn\sqrt{n}$

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13) $\sqrt{16u^4v^3}$

$16 = 4^2$
$u^4 = (u^2)^2$
$v^3 = v^2 \cdot v$

$\sqrt{16u^4v^3} = \sqrt{16 \cdot (u^2)^2 \cdot v^2 \cdot v} = \sqrt{16} \cdot \sqrt{(u^2)^2} \cdot \sqrt{v^2} \cdot \sqrt{v} = 4u^2v\sqrt{v}$

✔ Answer: $4u^2v\sqrt{v}$

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14) $\sqrt{28x^4y^3}$

Factor 28: $28 = 4 \cdot 7 = 2^2 \cdot 7$
$x^4 = (x^2)^2$
$y^3 = y^2 \cdot y$

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

✔ Answer: $2x^2y\sqrt{7y}$

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## ✔ Final Answers:

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

Let me know if you’d like a printable version or want to check your work against these!