Simplify radicals worksheet with 21 problems focusing on not perfect squares.

Let's solve each of these radicals step by step. The goal is to simplify each square root expression by factoring out perfect squares from under the radical.

We use the rule:

$$
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
$$

and we look for the largest perfect square factor of the number under the radical.

---

1. $\sqrt{32}$

Factor: $32 = 16 \cdot 2$, and $16$ is a perfect square.
$$
\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}
$$

---

2. $\sqrt{18}$

$18 = 9 \cdot 2$, and $9$ is a perfect square.
$$
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
$$

---

3. $\sqrt{28}$

$28 = 4 \cdot 7$, $4$ is a perfect square.
$$
\sqrt{28} = \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}
$$

---

4. $\sqrt{40}$

$40 = 4 \cdot 10$, $4$ is a perfect square.
$$
\sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10}
$$

---

5. $\sqrt{48}$

$48 = 16 \cdot 3$, $16$ is a perfect square.
$$
\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}
$$

---

6. $\sqrt{x^2}$

This is a perfect square:
$$
\sqrt{x^2} = x \quad \text{(assuming } x \geq 0\text{)}
$$

---

7. $\sqrt{y^2}$

Same as above:
$$
\sqrt{y^2} = y \quad \text{(assuming } y \geq 0\text{)}
$$

---

8. $\sqrt{12}$

$12 = 4 \cdot 3$, $4$ is a perfect square.
$$
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
$$

---

9. $\sqrt{24}$

$24 = 4 \cdot 6$, $4$ is a perfect square.
$$
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}
$$

---

10. $\sqrt{125}$

$125 = 25 \cdot 5$, $25$ is a perfect square.
$$
\sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}
$$

---

11. $\sqrt{150}$

$150 = 25 \cdot 6$, $25$ is a perfect square.
$$
\sqrt{150} = \sqrt{25 \cdot 6} = \sqrt{25} \cdot \sqrt{6} = 5\sqrt{6}
$$

---

12. $\sqrt{48}$

Already did this (same as #5):
$$
\sqrt{48} = 4\sqrt{3}
$$

---

13. $\sqrt{50}$

$50 = 25 \cdot 2$, $25$ is a perfect square.
$$
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
$$

---

14. $\sqrt{72}$

$72 = 36 \cdot 2$, $36$ is a perfect square.
$$
\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}
$$

---

15. $\sqrt{175}$

$175 = 25 \cdot 7$, $25$ is a perfect square.
$$
\sqrt{175} = \sqrt{25 \cdot 7} = \sqrt{25} \cdot \sqrt{7} = 5\sqrt{7}
$$

---

16. $\sqrt{80}$

$80 = 16 \cdot 5$, $16$ is a perfect square.
$$
\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5}
$$

---

17. $\sqrt{48}$

Again, same as #5 and #12:
$$
\sqrt{48} = 4\sqrt{3}
$$

---

18. $\sqrt{120}$

$120 = 4 \cdot 30$, but $30$ is not simplified. Try better:
$120 = 4 \cdot 30 = 4 \cdot (2 \cdot 15) = 4 \cdot 2 \cdot 15$ — still not ideal.
Better: $120 = 4 \cdot 30$, but $30$ has no perfect square factors.
Wait: $120 = 4 \cdot 30$, so:
$$
\sqrt{120} = \sqrt{4 \cdot 30} = 2\sqrt{30}
$$

Alternatively, check: $120 = 4 \cdot 30$, yes. But can we do better?
No — $30 = 2 \cdot 3 \cdot 5$, no perfect square. So:
$$
\sqrt{120} = 2\sqrt{30}
$$

---

19. $\sqrt{108}$

$108 = 36 \cdot 3$, $36$ is a perfect square.
$$
\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3}
$$

---

20. $\sqrt{90}$

$90 = 9 \cdot 10$, $9$ is a perfect square.
$$
\sqrt{90} = \sqrt{9 \cdot 10} = \sqrt{9} \cdot \sqrt{10} = 3\sqrt{10}
$$

---

21. $\sqrt{148}$

$148 = 4 \cdot 37$, $4$ is a perfect square, $37$ is prime.
$$
\sqrt{148} = \sqrt{4 \cdot 37} = \sqrt{4} \cdot \sqrt{37} = 2\sqrt{37}
$$

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✔ Final Answers:

| Problem | Simplified |
|--------|------------|
| 1. $\sqrt{32}$ | $4\sqrt{2}$ |
| 2. $\sqrt{18}$ | $3\sqrt{2}$ |
| 3. $\sqrt{28}$ | $2\sqrt{7}$ |
| 4. $\sqrt{40}$ | $2\sqrt{10}$ |
| 5. $\sqrt{48}$ | $4\sqrt{3}$ |
| 6. $\sqrt{x^2}$ | $x$ |
| 7. $\sqrt{y^2}$ | $y$ |
| 8. $\sqrt{12}$ | $2\sqrt{3}$ |
| 9. $\sqrt{24}$ | $2\sqrt{6}$ |
| 10. $\sqrt{125}$ | $5\sqrt{5}$ |
| 11. $\sqrt{150}$ | $5\sqrt{6}$ |
| 12. $\sqrt{48}$ | $4\sqrt{3}$ |
| 13. $\sqrt{50}$ | $5\sqrt{2}$ |
| 14. $\sqrt{72}$ | $6\sqrt{2}$ |
| 15. $\sqrt{175}$ | $5\sqrt{7}$ |
| 16. $\sqrt{80}$ | $4\sqrt{5}$ |
| 17. $\sqrt{48}$ | $4\sqrt{3}$ |
| 18. $\sqrt{120}$ | $2\sqrt{30}$ |
| 19. $\sqrt{108}$ | $6\sqrt{3}$ |
| 20. $\sqrt{90}$ | $3\sqrt{10}$ |
| 21. $\sqrt{148}$ | $2\sqrt{37}$ |

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🔍 Summary Tips:

- Always look for the largest perfect square factor.
- Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, ...
- For variables like $x^2$, the square root is $x$ (if $x \geq 0$).
- If you're unsure, factor into primes and group pairs.

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