Simplifying radical expressions :: Algebra Helper - Free Printable

Let's solve each of these square root expressions step by step, simplifying them as much as possible. We'll use the property:

$$
\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}, \quad \text{and} \quad \sqrt{a^2} = a \text{ (for } a \geq 0\text{)}
$$

We'll also use exponent rules:
$$
\sqrt{a^n} = a^{n/2}, \quad \text{so } \sqrt{a^{2k}} = a^k, \quad \text{and } \sqrt{a^{2k+1}} = a^k \sqrt{a}
$$

---

1. $\sqrt{18}$

$$
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
$$

✔ Answer: $3\sqrt{2}$

---

2. $\sqrt{125}$

$$
\sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}
$$

✔ Answer: $5\sqrt{5}$

---

3. $\sqrt{72}$

$$
\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}
$$

✔ Answer: $6\sqrt{2}$

---

4. $\sqrt{180}$

$$
\sqrt{180} = \sqrt{36 \cdot 5} = \sqrt{36} \cdot \sqrt{5} = 6\sqrt{5}
$$

✔ Answer: $6\sqrt{5}$

---

5. $\sqrt{a^3}$

$$
\sqrt{a^3} = \sqrt{a^2 \cdot a} = \sqrt{a^2} \cdot \sqrt{a} = a\sqrt{a}
$$

✔ Answer: $a\sqrt{a}$

---

6. $\sqrt{b^7}$

$$
\sqrt{b^7} = \sqrt{b^6 \cdot b} = \sqrt{b^6} \cdot \sqrt{b} = b^3\sqrt{b}
$$

✔ Answer: $b^3\sqrt{b}$

---

7. $\sqrt{m^{11}}$

$$
\sqrt{m^{11}} = \sqrt{m^{10} \cdot m} = \sqrt{m^{10}} \cdot \sqrt{m} = m^5\sqrt{m}
$$

✔ Answer: $m^5\sqrt{m}$

---

8. $\sqrt{75x^7y^5}$

Break into parts:

- $\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}$
- $\sqrt{x^7} = \sqrt{x^6 \cdot x} = x^3\sqrt{x}$
- $\sqrt{y^5} = \sqrt{y^4 \cdot y} = y^2\sqrt{y}$

Now combine:

$$
\sqrt{75x^7y^5} = 5\sqrt{3} \cdot x^3\sqrt{x} \cdot y^2\sqrt{y} = 5x^3y^2 \sqrt{3xy}
$$

✔ Answer: $5x^3y^2\sqrt{3xy}$

---

9. $\sqrt{27a^{11}b^7}$

- $\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}$
- $\sqrt{a^{11}} = a^5\sqrt{a}$
- $\sqrt{b^7} = b^3\sqrt{b}$

Combine:

$$
= 3\sqrt{3} \cdot a^5\sqrt{a} \cdot b^3\sqrt{b} = 3a^5b^3 \sqrt{3ab}
$$

✔ Answer: $3a^5b^3\sqrt{3ab}$

---

10. $\sqrt{32a^7b^4}$

- $\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$
- $\sqrt{a^7} = a^3\sqrt{a}$
- $\sqrt{b^4} = b^2$

Combine:

$$
= 4\sqrt{2} \cdot a^3\sqrt{a} \cdot b^2 = 4a^3b^2 \sqrt{2a}
$$

✔ Answer: $4a^3b^2\sqrt{2a}$

---

11. $\sqrt{9a^8}$

- $\sqrt{9} = 3$
- $\sqrt{a^8} = a^4$

So:

$$
\sqrt{9a^8} = 3a^4
$$

✔ Answer: $3a^4$

---

12. $\sqrt{45a^7}$

- $\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}$
- $\sqrt{a^7} = a^3\sqrt{a}$

$$
= 3\sqrt{5} \cdot a^3\sqrt{a} = 3a^3\sqrt{5a}
$$

✔ Answer: $3a^3\sqrt{5a}$

---

13. $\sqrt{36x^2y^6}$

- $\sqrt{36} = 6$
- $\sqrt{x^2} = x$
- $\sqrt{y^6} = y^3$

$$
= 6xy^3
$$

✔ Answer: $6xy^3$

---

14. $\sqrt{12x^{20}y^8}$

- $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$
- $\sqrt{x^{20}} = x^{10}$
- $\sqrt{y^8} = y^4$

$$
= 2\sqrt{3} \cdot x^{10} \cdot y^4 = 2x^{10}y^4\sqrt{3}
$$

✔ Answer: $2x^{10}y^4\sqrt{3}$

---

15. $-\sqrt{200}$

First simplify $\sqrt{200}$:

$$
\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}
$$

So:

$$
-\sqrt{200} = -10\sqrt{2}
$$

✔ Answer: $-10\sqrt{2}$

---

16. $\sqrt{196}$

$$
\sqrt{196} = 14 \quad (\text{since } 14^2 = 196)
$$

✔ Answer: $14$

---

17. $\sqrt{63x^4y}$

- $\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}$
- $\sqrt{x^4} = x^2$
- $\sqrt{y} = \sqrt{y}$

$$
= 3\sqrt{7} \cdot x^2 \cdot \sqrt{y} = 3x^2\sqrt{7y}
$$

✔ Answer: $3x^2\sqrt{7y}$

---

18. $\sqrt{6x^3}$

- $\sqrt{6}$ stays
- $\sqrt{x^3} = x\sqrt{x}$

$$
= \sqrt{6} \cdot x\sqrt{x} = x\sqrt{6x}
$$

✔ Answer: $x\sqrt{6x}$

---

19. $\sqrt{100x^5y}$

- $\sqrt{100} = 10$
- $\sqrt{x^5} = x^2\sqrt{x}$
- $\sqrt{y} = \sqrt{y}$

$$
= 10 \cdot x^2\sqrt{x} \cdot \sqrt{y} = 10x^2\sqrt{xy}
$$

✔ Answer: $10x^2\sqrt{xy}$

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20. $\sqrt{80x^{100}y^{49}}$

- $\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}$
- $\sqrt{x^{100}} = x^{50}$
- $\sqrt{y^{49}} = \sqrt{y^{48} \cdot y} = y^{24}\sqrt{y}$

$$
= 4\sqrt{5} \cdot x^{50} \cdot y^{24}\sqrt{y} = 4x^{50}y^{24}\sqrt{5y}
$$

✔ Answer: $4x^{50}y^{24}\sqrt{5y}$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1. $\sqrt{18}$ | $3\sqrt{2}$ |
| 2. $\sqrt{125}$ | $5\sqrt{5}$ |
| 3. $\sqrt{72}$ | $6\sqrt{2}$ |
| 4. $\sqrt{180}$ | $6\sqrt{5}$ |
| 5. $\sqrt{a^3}$ | $a\sqrt{a}$ |
| 6. $\sqrt{b^7}$ | $b^3\sqrt{b}$ |
| 7. $\sqrt{m^{11}}$ | $m^5\sqrt{m}$ |
| 8. $\sqrt{75x^7y^5}$ | $5x^3y^2\sqrt{3xy}$ |
| 9. $\sqrt{27a^{11}b^7}$ | $3a^5b^3\sqrt{3ab}$ |
| 10. $\sqrt{32a^7b^4}$ | $4a^3b^2\sqrt{2a}$ |
| 11. $\sqrt{9a^8}$ | $3a^4$ |
| 12. $\sqrt{45a^7}$ | $3a^3\sqrt{5a}$ |
| 13. $\sqrt{36x^2y^6}$ | $6xy^3$ |
| 14. $\sqrt{12x^{20}y^8}$ | $2x^{10}y^4\sqrt{3}$ |
| 15. $-\sqrt{200}$ | $-10\sqrt{2}$ |
| 16. $\sqrt{196}$ | $14$ |
| 17. $\sqrt{63x^4y}$ | $3x^2\sqrt{7y}$ |
| 18. $\sqrt{6x^3}$ | $x\sqrt{6x}$ |
| 19. $\sqrt{100x^5y}$ | $10x^2\sqrt{xy}$ |
| 20. $\sqrt{80x^{100}y^{49}}$ | $4x^{50}y^{24}\sqrt{5y}$ |

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