Concept 14: Square Roots - Free Printable

Let's solve each of the square root simplification problems step by step. I'll show all the work and explain the process.

---

1. √132

We want to simplify √132.

- Factor 132:
132 = 4 × 33
(Note: 4 is a perfect square)

So:
$$
\sqrt{132} = \sqrt{4 \times 33} = \sqrt{4} \cdot \sqrt{33} = 2\sqrt{33}
$$

✔ Answer: $ 2\sqrt{33} $

---

2. √75

Factor 75:
75 = 25 × 3
25 is a perfect square.

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

✔ Answer: $ 5\sqrt{3} $

---

3. √80 + √20

Simplify each term:

- √80 = √(16 × 5) = √16 × √5 = 4√5
- √20 = √(4 × 5) = √4 × √5 = 2√5

Now add:
$$
4\sqrt{5} + 2\sqrt{5} = (4+2)\sqrt{5} = 6\sqrt{5}
$$

✔ Answer: $ 6\sqrt{5} $

---

4. √2 + √40

Simplify √40:
√40 = √(4 × 10) = 2√10

So:
$$
\sqrt{2} + 2\sqrt{10}
$$

These are not like terms (different radicals), so we cannot combine them.

✔ Answer: $ \sqrt{2} + 2\sqrt{10} $

---

5. √54 + √15

Simplify each:

- √54 = √(9 × 6) = 3√6
- √15 cannot be simplified

So:
$$
3\sqrt{6} + \sqrt{15}
$$

Not like terms → cannot combine.

✔ Answer: $ 3\sqrt{6} + \sqrt{15} $

---

6. √5 + √90

Simplify √90:
√90 = √(9 × 10) = 3√10

So:
$$
\sqrt{5} + 3\sqrt{10}
$$

Not like terms → can't combine.

✔ Answer: $ \sqrt{5} + 3\sqrt{10} $

---

7. √100 / √625

This is a fraction of square roots:
$$
\frac{\sqrt{100}}{\sqrt{625}} = \sqrt{\frac{100}{625}} = \sqrt{\frac{4}{25}} = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5}
$$

Alternatively, just compute:
√100 = 10, √625 = 25 → 10/25 = 2/5

✔ Answer: $ \frac{2}{5} $

---

8. √(3/36)

Simplify inside:
$$
\frac{3}{36} = \frac{1}{12}
$$

So:
$$
\sqrt{\frac{1}{12}} = \frac{\sqrt{1}}{\sqrt{12}} = \frac{1}{\sqrt{12}}
$$

Simplify √12 = √(4×3) = 2√3

So:
$$
\frac{1}{2\sqrt{3}}
$$

Rationalize the denominator:
$$
\frac{1}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{6}
$$

✔ Answer: $ \frac{\sqrt{3}}{6} $

---

9. √(225/24)

Simplify:
$$
\frac{225}{24} = \frac{75}{8} \quad \text{(divide numerator and denominator by 3)}
$$

But better to write:
$$
\sqrt{\frac{225}{24}} = \frac{\sqrt{225}}{\sqrt{24}} = \frac{15}{\sqrt{24}}
$$

Simplify √24 = √(4×6) = 2√6

So:
$$
\frac{15}{2\sqrt{6}}
$$

Rationalize:
$$
\frac{15}{2\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{15\sqrt{6}}{2 \cdot 6} = \frac{15\sqrt{6}}{12} = \frac{5\sqrt{6}}{4}
$$

✔ Answer: $ \frac{5\sqrt{6}}{4} $

---

10. √(128/90)

Simplify the fraction first:
$$
\frac{128}{90} = \frac{64}{45} \quad \text{(divide numerator and denominator by 2)}
$$

Now:
$$
\sqrt{\frac{64}{45}} = \frac{\sqrt{64}}{\sqrt{45}} = \frac{8}{\sqrt{45}}
$$

Simplify √45 = √(9×5) = 3√5

So:
$$
\frac{8}{3\sqrt{5}}
$$

Rationalize:
$$
\frac{8}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{8\sqrt{5}}{3 \cdot 5} = \frac{8\sqrt{5}}{15}
$$

✔ Answer: $ \frac{8\sqrt{5}}{15} $

---

11. √(16/400)

$$
\frac{16}{400} = \frac{1}{25}
$$

So:
$$
\sqrt{\frac{1}{25}} = \frac{1}{5}
$$

✔ Answer: $ \frac{1}{5} $

---

12. √(7/12)

$$
\sqrt{\frac{7}{12}} = \frac{\sqrt{7}}{\sqrt{12}} = \frac{\sqrt{7}}{2\sqrt{3}}
$$

Rationalize:
$$
\frac{\sqrt{7}}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{21}}{2 \cdot 3} = \frac{\sqrt{21}}{6}
$$

✔ Answer: $ \frac{\sqrt{21}}{6} $

---

13. √(11/75)

$$
\sqrt{\frac{11}{75}} = \frac{\sqrt{11}}{\sqrt{75}} = \frac{\sqrt{11}}{5\sqrt{3}}
$$

Rationalize:
$$
\frac{\sqrt{11}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{33}}{5 \cdot 3} = \frac{\sqrt{33}}{15}
$$

✔ Answer: $ \frac{\sqrt{33}}{15} $

---

14. √(5/18)

$$
\sqrt{\frac{5}{18}} = \frac{\sqrt{5}}{\sqrt{18}} = \frac{\sqrt{5}}{3\sqrt{2}}
$$

Rationalize:
$$
\frac{\sqrt{5}}{3\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{10}}{3 \cdot 2} = \frac{\sqrt{10}}{6}
$$

✔ Answer: $ \frac{\sqrt{10}}{6} $

---

15. √(25/15)

Simplify the fraction:
$$
\frac{25}{15} = \frac{5}{3}
$$

So:
$$
\sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}} = \frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{15}}{3}
$$

✔ Answer: $ \frac{\sqrt{15}}{3} $

---

✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1. √132 | $ 2\sqrt{33} $ |
| 2. √75 | $ 5\sqrt{3} $ |
| 3. √80 + √20 | $ 6\sqrt{5} $ |
| 4. √2 + √40 | $ \sqrt{2} + 2\sqrt{10} $ |
| 5. √54 + √15 | $ 3\sqrt{6} + \sqrt{15} $ |
| 6. √5 + √90 | $ \sqrt{5} + 3\sqrt{10} $ |
| 7. √100 / √625 | $ \frac{2}{5} $ |
| 8. √(3/36) | $ \frac{\sqrt{3}}{6} $ |
| 9. √(225/24) | $ \frac{5\sqrt{6}}{4} $ |
| 10. √(128/90) | $ \frac{8\sqrt{5}}{15} $ |
| 11. √(16/400) | $ \frac{1}{5} $ |
| 12. √(7/12) | $ \frac{\sqrt{21}}{6} $ |
| 13. √(11/75) | $ \frac{\sqrt{33}}{15} $ |
| 14. √(5/18) | $ \frac{\sqrt{10}}{6} $ |
| 15. √(25/15) | $ \frac{\sqrt{15}}{3} $ |

Let me know if you'd like this formatted as a printable answer sheet!