Simplify Radical Equations Worksheet - Free Printable

To solve the given problems involving simplifying radical expressions, we need to factorize the numbers under the square roots and simplify them step by step. Let's go through each problem:

---

1) \(\sqrt{125n}\)

- Factorize 125:
\[
125 = 5 \times 5 \times 5 = 5^3
\]
- Rewrite the expression:
\[
\sqrt{125n} = \sqrt{5^3 \cdot n}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{5^3 \cdot n} = \sqrt{5^2 \cdot 5 \cdot n} = 5\sqrt{5n}
\]

Answer:
\[
\boxed{5\sqrt{5n}}
\]

---

2) \(\sqrt{216v}\)

- Factorize 216:
\[
216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3
\]
- Rewrite the expression:
\[
\sqrt{216v} = \sqrt{2^3 \cdot 3^3 \cdot v}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^3 \cdot 3^3 \cdot v} = \sqrt{(2^2 \cdot 3^2) \cdot (2 \cdot 3 \cdot v)} = 2 \cdot 3 \cdot \sqrt{2 \cdot 3 \cdot v} = 6\sqrt{6v}
\]

Answer:
\[
\boxed{6\sqrt{6v}}
\]

---

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

- Factorize 512:
\[
512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9
\]
- Rewrite the expression:
\[
\sqrt{512k^2} = \sqrt{2^9 \cdot k^2}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^9 \cdot k^2} = \sqrt{(2^8) \cdot (2) \cdot k^2} = 2^4 \cdot k \cdot \sqrt{2} = 16k\sqrt{2}
\]

Answer:
\[
\boxed{16k\sqrt{2}}
\]

---

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

- Factorize 512:
\[
512 = 2^9
\]
- Rewrite the expression:
\[
\sqrt{512m^3} = \sqrt{2^9 \cdot m^3}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^9 \cdot m^3} = \sqrt{(2^8) \cdot (2) \cdot (m^2) \cdot m} = 2^4 \cdot m \cdot \sqrt{2m} = 16m\sqrt{2m}
\]

Answer:
\[
\boxed{16m\sqrt{2m}}
\]

---

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

- Factorize 216:
\[
216 = 2^3 \times 3^3
\]
- Rewrite the expression:
\[
\sqrt{216k^4} = \sqrt{2^3 \cdot 3^3 \cdot k^4}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^3 \cdot 3^3 \cdot k^4} = \sqrt{(2^2 \cdot 3^2) \cdot (2 \cdot 3) \cdot k^4} = 2 \cdot 3 \cdot k^2 \cdot \sqrt{6} = 6k^2\sqrt{6}
\]

Answer:
\[
\boxed{6k^2\sqrt{6}}
\]

---

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

- Factorize 100:
\[
100 = 10^2 = 2^2 \times 5^2
\]
- Rewrite the expression:
\[
\sqrt{100v^3} = \sqrt{2^2 \cdot 5^2 \cdot v^3}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^2 \cdot 5^2 \cdot v^3} = \sqrt{(2^2 \cdot 5^2) \cdot (v^2) \cdot v} = 2 \cdot 5 \cdot v \cdot \sqrt{v} = 10v\sqrt{v}
\]

Answer:
\[
\boxed{10v\sqrt{v}}
\]

---

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

- Factorize 80:
\[
80 = 2^4 \times 5
\]
- Rewrite the expression:
\[
\sqrt{80p^3} = \sqrt{2^4 \cdot 5 \cdot p^3}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^4 \cdot 5 \cdot p^3} = \sqrt{(2^4) \cdot (p^2) \cdot (5 \cdot p)} = 2^2 \cdot p \cdot \sqrt{5p} = 4p\sqrt{5p}
\]

Answer:
\[
\boxed{4p\sqrt{5p}}
\]

---

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

- Factorize 45:
\[
45 = 3^2 \times 5
\]
- Rewrite the expression:
\[
\sqrt{45p^2} = \sqrt{3^2 \cdot 5 \cdot p^2}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{3^2 \cdot 5 \cdot p^2} = \sqrt{(3^2) \cdot (p^2) \cdot 5} = 3 \cdot p \cdot \sqrt{5} = 3p\sqrt{5}
\]

Answer:
\[
\boxed{3p\sqrt{5}}
\]

---

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

- Factorize 147:
\[
147 = 3 \times 7 \times 7 = 3 \times 7^2
\]
- Rewrite the expression:
\[
\sqrt{147m^3} = \sqrt{3 \cdot 7^2 \cdot m^3}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{3 \cdot 7^2 \cdot m^3} = \sqrt{(7^2) \cdot (m^2) \cdot (3 \cdot m)} = 7 \cdot m \cdot \sqrt{3m} = 7m\sqrt{3m}
\]

Answer:
\[
\boxed{7m\sqrt{3m}}
\]

---

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

- Factorize 100:
\[
100 = 10^2 = 2^2 \times 5^2
\]
- Rewrite the expression:
\[
\sqrt{100m^4} = \sqrt{2^2 \cdot 5^2 \cdot m^4}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^2 \cdot 5^2 \cdot m^4} = \sqrt{(2^2) \cdot (5^2) \cdot (m^4)} = 2 \cdot 5 \cdot m^2 = 10m^2
\]

Answer:
\[
\boxed{10m^2}
\]

---

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

- Factorize 75:
\[
75 = 3 \times 5^2
\]
- Rewrite the expression:
\[
\sqrt{75x^2} = \sqrt{3 \cdot 5^2 \cdot x^2}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{3 \cdot 5^2 \cdot x^2} = \sqrt{(5^2) \cdot (x^2) \cdot 3} = 5 \cdot x \cdot \sqrt{3} = 5x\sqrt{3}
\]

Answer:
\[
\boxed{5x\sqrt{3}}
\]

---

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

- Factorize 64:
\[
64 = 8^2 = 2^6 = (2^3)^2
\]
- Rewrite the expression:
\[
\sqrt{64m^4} = \sqrt{2^6 \cdot m^4}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^6 \cdot m^4} = \sqrt{(2^6) \cdot (m^4)} = 2^3 \cdot m^2 = 8m^2
\]

Answer:
\[
\boxed{8m^2}
\]

---

13) \(\sqrt{16u^2}\)

- Factorize 16:
\[
16 = 4^2 = 2^4
\]
- Rewrite the expression:
\[
\sqrt{16u^2} = \sqrt{2^4 \cdot u^2}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^4 \cdot u^2} = \sqrt{(2^4) \cdot (u^2)} = 2^2 \cdot u = 4u
\]

Answer:
\[
\boxed{4u}
\]

---

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

- Factorize 28:
\[
28 = 2^2 \times 7
\]
- Rewrite the expression:
\[
\sqrt{28x^3} = \sqrt{2^2 \cdot 7 \cdot x^3}
\]
- Simplify using the property \(\sqrt{a^2} = a\):
\[
\sqrt{2^2 \cdot 7 \cdot x^3} = \sqrt{(2^2) \cdot (x^2) \cdot (7 \cdot x)} = 2 \cdot x \cdot \sqrt{7x} = 2x\sqrt{7x}
\]

Answer:
\[
\boxed{2x\sqrt{7x}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1) & \ 5\sqrt{5n} \\
2) & \ 6\sqrt{6v} \\
3) & \ 16k\sqrt{2} \\
4) & \ 16m\sqrt{2m} \\
5) & \ 6k^2\sqrt{6} \\
6) & \ 10v\sqrt{v} \\
7) & \ 4p\sqrt{5p} \\
8) & \ 3p\sqrt{5} \\
9) & \ 7m\sqrt{3m} \\
10) & \ 10m^2 \\
11) & \ 5x\sqrt{3} \\
12) & \ 8m^2 \\
13) & \ 4u \\
14) & \ 2x\sqrt{7x}
\end{aligned}
}
\]