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.

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

---

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: $5\sqrt{5n}$

---

2) $\sqrt{216v}$

Factor 216:
$216 = 36 \times 6 = 6^2 \times 6$, or better:
$216 = 36 \times 6 = (6^2) \cdot 6$

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

✔ Answer: $6\sqrt{6v}$

---

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

Factor 512:
$512 = 256 \times 2 = 16^2 \times 2$, or $512 = 2^9$, but easier:

$512 = 256 \cdot 2 = (16^2) \cdot 2$

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

✔ Answer: $16k\sqrt{2}$

---

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

We already know $512 = 256 \cdot 2 = 16^2 \cdot 2$

And $m^3 = m^2 \cdot m$

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

✔ Answer: $16m\sqrt{2m}$

---

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

We had $216 = 36 \cdot 6 = 6^2 \cdot 6$

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

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

✔ Answer: $6k^2\sqrt{6}$

---

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

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

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

✔ Answer: $10v\sqrt{v}$

---

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

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

$p^3 = p^2 \cdot p$

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: $4p\sqrt{5p}$

---

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

$45 = 9 \cdot 5 = 3^2 \cdot 5$

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

✔ Answer: $3p\sqrt{5}$

---

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

$147 = 49 \cdot 3 = 7^2 \cdot 3$

$m^3 = m^2 \cdot m$

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

✔ Answer: $7m\sqrt{3m}$

---

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

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

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

✔ Answer: $10m^2$

---

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

$75 = 25 \cdot 3 = 5^2 \cdot 3$

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

✔ Answer: $5x\sqrt{3}$

---

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

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

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

✔ Answer: $8m^2$

---

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: $4u^2$

---

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

$28 = 4 \cdot 7 = 2^2 \cdot 7$

$x^3 = x^2 \cdot x$

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

✔ Answer: $2x\sqrt{7x}$

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✔ Final Answers:

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) $2x\sqrt{7x}$

Let me know if you'd like a printable version or explanation for any specific one!