Simplifying Radicals of Index 2 (Assume all variables are positive.) - Free Printable

Problem: Simplifying Radicals of Index 2

The task is to simplify the given square roots (radicals of index 2). We will use the properties of square roots and factorization to simplify each expression. Let's go through the problems step by step.

---

#### 1. Simplify \( \sqrt{125} \)

- Factorize 125 into prime factors:
\[
125 = 5 \times 5 \times 5 = 5^3
\]
- Rewrite the square root:
\[
\sqrt{125} = \sqrt{5^3} = \sqrt{5^2 \cdot 5} = \sqrt{5^2} \cdot \sqrt{5} = 5\sqrt{5}
\]

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

---

#### 2. Simplify \( \sqrt{72} \)

- Factorize 72 into prime factors:
\[
72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2
\]
- Rewrite the square root:
\[
\sqrt{72} = \sqrt{2^3 \cdot 3^2} = \sqrt{2^2 \cdot 2 \cdot 3^2} = \sqrt{2^2} \cdot \sqrt{3^2} \cdot \sqrt{2} = 2 \cdot 3 \cdot \sqrt{2} = 6\sqrt{2}
\]

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

---

#### 3. Simplify \( \sqrt{63} \)

- Factorize 63 into prime factors:
\[
63 = 3 \times 3 \times 7 = 3^2 \times 7
\]
- Rewrite the square root:
\[
\sqrt{63} = \sqrt{3^2 \cdot 7} = \sqrt{3^2} \cdot \sqrt{7} = 3\sqrt{7}
\]

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

---

#### 4. Simplify \( \sqrt{45} \)

- Factorize 45 into prime factors:
\[
45 = 3 \times 3 \times 5 = 3^2 \times 5
\]
- Rewrite the square root:
\[
\sqrt{45} = \sqrt{3^2 \cdot 5} = \sqrt{3^2} \cdot \sqrt{5} = 3\sqrt{5}
\]

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

---

#### 5. Simplify \( \sqrt{12} \)

- Factorize 12 into prime factors:
\[
12 = 2 \times 2 \times 3 = 2^2 \times 3
\]
- Rewrite the square root:
\[
\sqrt{12} = \sqrt{2^2 \cdot 3} = \sqrt{2^2} \cdot \sqrt{3} = 2\sqrt{3}
\]

Answer:
\[
\boxed{2\sqrt{3}}
\]

---

#### 6. Simplify \( \sqrt{18} \)

- Factorize 18 into prime factors:
\[
18 = 2 \times 3 \times 3 = 2 \times 3^2
\]
- Rewrite the square root:
\[
\sqrt{18} = \sqrt{2 \cdot 3^2} = \sqrt{3^2} \cdot \sqrt{2} = 3\sqrt{2}
\]

Answer:
\[
\boxed{3\sqrt{2}}
\]

---

#### 7. Simplify \( \sqrt{96} \)

- Factorize 96 into prime factors:
\[
96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^5 \times 3
\]
- Rewrite the square root:
\[
\sqrt{96} = \sqrt{2^5 \cdot 3} = \sqrt{2^4 \cdot 2 \cdot 3} = \sqrt{2^4} \cdot \sqrt{2 \cdot 3} = 2^2 \cdot \sqrt{6} = 4\sqrt{6}
\]

Answer:
\[
\boxed{4\sqrt{6}}
\]

---

#### 8. Simplify \( \sqrt{150} \)

- Factorize 150 into prime factors:
\[
150 = 2 \times 3 \times 5 \times 5 = 2 \times 3 \times 5^2
\]
- Rewrite the square root:
\[
\sqrt{150} = \sqrt{2 \cdot 3 \cdot 5^2} = \sqrt{5^2} \cdot \sqrt{2 \cdot 3} = 5\sqrt{6}
\]

Answer:
\[
\boxed{5\sqrt{6}}
\]

---

#### 9. Simplify \( \sqrt{112} \)

- Factorize 112 into prime factors:
\[
112 = 2 \times 2 \times 2 \times 2 \times 7 = 2^4 \times 7
\]
- Rewrite the square root:
\[
\sqrt{112} = \sqrt{2^4 \cdot 7} = \sqrt{2^4} \cdot \sqrt{7} = 2^2 \cdot \sqrt{7} = 4\sqrt{7}
\]

Answer:
\[
\boxed{4\sqrt{7}}
\]

---

#### 10. Simplify \( \sqrt{50} \)

- Factorize 50 into prime factors:
\[
50 = 2 \times 5 \times 5 = 2 \times 5^2
\]
- Rewrite the square root:
\[
\sqrt{50} = \sqrt{2 \cdot 5^2} = \sqrt{5^2} \cdot \sqrt{2} = 5\sqrt{2}
\]

Answer:
\[
\boxed{5\sqrt{2}}
\]

---

#### 11. Simplify \( \sqrt{8a^4} \)

- Factorize the expression:
\[
8a^4 = 2 \times 2 \times 2 \times a^4 = 2^3 \cdot a^4
\]
- Rewrite the square root:
\[
\sqrt{8a^4} = \sqrt{2^3 \cdot a^4} = \sqrt{2^2 \cdot 2 \cdot a^4} = \sqrt{2^2} \cdot \sqrt{a^4} \cdot \sqrt{2} = 2 \cdot a^2 \cdot \sqrt{2} = 2a^2\sqrt{2}
\]

Answer:
\[
\boxed{2a^2\sqrt{2}}
\]

---

#### 12. Simplify \( \sqrt{200x} \)

- Factorize the expression:
\[
200x = 2 \times 2 \times 2 \times 5 \times 5 \times x = 2^3 \cdot 5^2 \cdot x
\]
- Rewrite the square root:
\[
\sqrt{200x} = \sqrt{2^3 \cdot 5^2 \cdot x} = \sqrt{2^2 \cdot 2 \cdot 5^2 \cdot x} = \sqrt{2^2} \cdot \sqrt{5^2} \cdot \sqrt{2x} = 2 \cdot 5 \cdot \sqrt{2x} = 10\sqrt{2x}
\]

Answer:
\[
\boxed{10\sqrt{2x}}
\]

---

#### 13. Simplify \( \sqrt{20x^3} \)

- Factorize the expression:
\[
20x^3 = 2 \times 2 \times 5 \times x \times x \times x = 2^2 \cdot 5 \cdot x^3
\]
- Rewrite the square root:
\[
\sqrt{20x^3} = \sqrt{2^2 \cdot 5 \cdot x^3} = \sqrt{2^2 \cdot x^2 \cdot x \cdot 5} = \sqrt{2^2} \cdot \sqrt{x^2} \cdot \sqrt{5x} = 2 \cdot x \cdot \sqrt{5x} = 2x\sqrt{5x}
\]

Answer:
\[
\boxed{2x\sqrt{5x}}
\]

---

#### 14. Simplify \( \sqrt{45m} \)

- Factorize the expression:
\[
45m = 3 \times 3 \times 5 \times m = 3^2 \cdot 5 \cdot m
\]
- Rewrite the square root:
\[
\sqrt{45m} = \sqrt{3^2 \cdot 5 \cdot m} = \sqrt{3^2} \cdot \sqrt{5m} = 3\sqrt{5m}
\]

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

---

#### 15. Simplify \( \sqrt{98x^2} \)

- Factorize the expression:
\[
98x^2 = 2 \times 7 \times 7 \times x^2 = 2 \cdot 7^2 \cdot x^2
\]
- Rewrite the square root:
\[
\sqrt{98x^2} = \sqrt{2 \cdot 7^2 \cdot x^2} = \sqrt{7^2} \cdot \sqrt{x^2} \cdot \sqrt{2} = 7 \cdot x \cdot \sqrt{2} = 7x\sqrt{2}
\]

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

---

#### 16. Simplify \( \sqrt{24a^4} \)

- Factorize the expression:
\[
24a^4 = 2 \times 2 \times 2 \times 3 \times a^4 = 2^3 \cdot 3 \cdot a^4
\]
- Rewrite the square root:
\[
\sqrt{24a^4} = \sqrt{2^3 \cdot 3 \cdot a^4} = \sqrt{2^2 \cdot 2 \cdot 3 \cdot a^4} = \sqrt{2^2} \cdot \sqrt{a^4} \cdot \sqrt{2 \cdot 3} = 2 \cdot a^2 \cdot \sqrt{6} = 2a^2\sqrt{6}
\]

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

---

#### 17. Simplify \( \sqrt{75x^2} \)

- Factorize the expression:
\[
75x^2 = 3 \times 5 \times 5 \times x^2 = 3 \cdot 5^2 \cdot x^2
\]
- Rewrite the square root:
\[
\sqrt{75x^2} = \sqrt{3 \cdot 5^2 \cdot x^2} = \sqrt{5^2} \cdot \sqrt{x^2} \cdot \sqrt{3} = 5 \cdot x \cdot \sqrt{3} = 5x\sqrt{3}
\]

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

---

#### 18. Simplify \( \sqrt{32k^8} \)

- Factorize the expression:
\[
32k^8 = 2 \times 2 \times 2 \times 2 \times 2 \times k^8 = 2^5 \cdot k^8
\]
- Rewrite the square root:
\[
\sqrt{32k^8} = \sqrt{2^5 \cdot k^8} = \sqrt{2^4 \cdot 2 \cdot k^8} = \sqrt{2^4} \cdot \sqrt{k^8} \cdot \sqrt{2} = 2^2 \cdot k^4 \cdot \sqrt{2} = 4k^4\sqrt{2}
\]

Answer:
\[
\boxed{4k^4\sqrt{2}}
\]

---

#### 19. Simplify \( \sqrt{98m^{12}} \)

- Factorize the expression:
\[
98m^{12} = 2 \times 7 \times 7 \times m^{12} = 2 \cdot 7^2 \cdot m^{12}
\]
- Rewrite the square root:
\[
\sqrt{98m^{12}} = \sqrt{2 \cdot 7^2 \cdot m^{12}} = \sqrt{7^2} \cdot \sqrt{m^{12}} \cdot \sqrt{2} = 7 \cdot m^6 \cdot \sqrt{2} = 7m^6\sqrt{2}
\]

Answer:
\[
\boxed{7m^6\sqrt{2}}
\]

---

#### 20. Simplify \( \sqrt{28k^8} \)

- Factorize the expression:
\[
28k^8 = 2 \times 2 \times 7 \times k^8 = 2^2 \cdot 7 \cdot k^8
\]
- Rewrite the square root:
\[
\sqrt{28k^8} = \sqrt{2^2 \cdot 7 \cdot k^8} = \sqrt{2^2} \cdot \sqrt{k^8} \cdot \sqrt{7} = 2 \cdot k^4 \cdot \sqrt{7} = 2k^4\sqrt{7}
\]

Answer:
\[
\boxed{2k^4\sqrt{7}}
\]

---

Final Answers for the First 20 Problems:

\[
\boxed{
\begin{aligned}
1. & \ 5\sqrt{5} \\
2. & \ 6\sqrt{2} \\
3. & \ 3\sqrt{7} \\
4. & \ 3\sqrt{5} \\
5. & \ 2\sqrt{3} \\
6. & \ 3\sqrt{2} \\
7. & \ 4\sqrt{6} \\
8. & \ 5\sqrt{6} \\
9. & \ 4\sqrt{7} \\
10. & \ 5\sqrt{2} \\
11. & \ 2a^2\sqrt{2} \\
12. & \ 10\sqrt{2x} \\
13. & \ 2x\sqrt{5x} \\
14. & \ 3\sqrt{5m} \\
15. & \ 7x\sqrt{2} \\
16. & \ 2a^2\sqrt{6} \\
17. & \ 5x\sqrt{3} \\
18. & \ 4k^4\sqrt{2} \\
19. & \ 7m^6\sqrt{2} \\
20. & \ 2k^4\sqrt{7}
\end{aligned}
}
\]