Simplify square roots practice worksheet for math students.

Let's simplify each of the square roots step by step. We'll look for perfect square factors of each number under the square root and then simplify.

---

1. √18

Break 18 into factors:
- 18 = 9 × 2
- 9 is a perfect square: √9 = 3

So:
$$
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
$$

✔ Answer: $ 3\sqrt{2} $

---

2. √27

Break 27 into factors:
- 27 = 9 × 3
- 9 is a perfect square: √9 = 3

So:
$$
\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
$$

✔ Answer: $ 3\sqrt{3} $

---

3. √8

Break 8 into factors:
- 8 = 4 × 2
- 4 is a perfect square: √4 = 2

So:
$$
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
$$

✔ Answer: $ 2\sqrt{2} $

---

4. √5

5 is a prime number and has no perfect square factors other than 1.

So:
$$
\sqrt{5} \text{ cannot be simplified further.}
$$

✔ Answer: $ \sqrt{5} $

---

5. √28

Break 28 into factors:
- 28 = 4 × 7
- 4 is a perfect square: √4 = 2

So:
$$
\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}
$$

✔ Answer: $ 2\sqrt{7} $

---

6. √75

Break 75 into factors:
- 75 = 25 × 3
- 25 is a perfect square: √25 = 5

So:
$$
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}
$$

✔ Answer: $ 5\sqrt{3} $

---

7. √45

Break 45 into factors:
- 45 = 9 × 5
- 9 is a perfect square: √9 = 3

So:
$$
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}
$$

✔ Answer: $ 3\sqrt{5} $

---

8. √32

Break 32 into factors:
- 32 = 16 × 2
- 16 is a perfect square: √16 = 4

So:
$$
\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}
$$

✔ Answer: $ 4\sqrt{2} $

---

9. √36

36 is a perfect square:
- √36 = 6

✔ Answer: $ 6 $

---

10. √128

Break 128 into factors:
- 128 = 64 × 2
- 64 is a perfect square: √64 = 8

So:
$$
\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \cdot \sqrt{2} = 8\sqrt{2}
$$

✔ Answer: $ 8\sqrt{2} $

---

✔ Final Answers:

| Problem | Simplified Form |
|--------|------------------|
| 1. √18 | $ 3\sqrt{2} $ |
| 2. √27 | $ 3\sqrt{3} $ |
| 3. √8 | $ 2\sqrt{2} $ |
| 4. √5 | $ \sqrt{5} $ |
| 5. √28 | $ 2\sqrt{7} $ |
| 6. √75 | $ 5\sqrt{3} $ |
| 7. √45 | $ 3\sqrt{5} $ |
| 8. √32 | $ 4\sqrt{2} $ |
| 9. √36 | $ 6 $ |
| 10. √128 | $ 8\sqrt{2} $ |

---

Key Strategy: Always look for the largest perfect square that divides the number under the square root, then use the property:
$$
\sqrt{a \times b} = \sqrt{a} \cdot \sqrt{b}
$$
and simplify accordingly.