Fun & Challenging Simplifying Radicals Worksheet - The Teach ... - Free Printable

It looks like you've uploaded a list of radical expression simplification problems (numbered 15 to 30). Let's go through each one step by step and simplify the expressions.

We'll use the following rules:

- $\sqrt{a^2} = a$ for $a \geq 0$
- $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
- $\sqrt{a^n} = a^{n/2}$ if $n$ is even
- Coefficients are factored out separately
- We factor perfect squares from under the radical

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15) $\sqrt{36x^2y^3}$

Break down:
- $36 = 6^2$, so $\sqrt{36} = 6$
- $x^2$: $\sqrt{x^2} = x$
- $y^3 = y^2 \cdot y$, so $\sqrt{y^3} = \sqrt{y^2 \cdot y} = y\sqrt{y}$

So:
$$
\sqrt{36x^2y^3} = 6 \cdot x \cdot y\sqrt{y} = 6xy\sqrt{y}
$$

✔ Answer: $6xy\sqrt{y}$

---

16) $\sqrt{384x^5y^3}$

Factor 384:
- $384 = 64 \times 6 = 8^2 \times 6$, so $\sqrt{384} = \sqrt{64 \cdot 6} = 8\sqrt{6}$

Now variables:
- $x^5 = x^4 \cdot x$, so $\sqrt{x^5} = \sqrt{x^4 \cdot x} = x^2\sqrt{x}$
- $y^3 = y^2 \cdot y$, so $\sqrt{y^3} = y\sqrt{y}$

Combine:
$$
\sqrt{384x^5y^3} = 8\sqrt{6} \cdot x^2\sqrt{x} \cdot y\sqrt{y} = 8x^2y\sqrt{6xy}
$$

✔ Answer: $8x^2y\sqrt{6xy}$

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17) $7\sqrt{96m^3}$

First, simplify $\sqrt{96m^3}$

- $96 = 16 \cdot 6 = 4^2 \cdot 6$, so $\sqrt{96} = 4\sqrt{6}$
- $m^3 = m^2 \cdot m$, so $\sqrt{m^3} = m\sqrt{m}$

So:
$$
\sqrt{96m^3} = 4\sqrt{6} \cdot m\sqrt{m} = 4m\sqrt{6m}
$$

Now multiply by 7:
$$
7 \cdot 4m\sqrt{6m} = 28m\sqrt{6m}
$$

✔ Answer: $28m\sqrt{6m}$

---

18) $6\sqrt{72x^2}$

Simplify $\sqrt{72x^2}$

- $72 = 36 \cdot 2 = 6^2 \cdot 2$, so $\sqrt{72} = 6\sqrt{2}$
- $\sqrt{x^2} = x$

So:
$$
\sqrt{72x^2} = 6x\sqrt{2}
$$

Multiply by 6:
$$
6 \cdot 6x\sqrt{2} = 36x\sqrt{2}
$$

✔ Answer: $36x\sqrt{2}$

---

19) $-6\sqrt{150r}$

Simplify $\sqrt{150r}$

- $150 = 25 \cdot 6 = 5^2 \cdot 6$, so $\sqrt{150} = 5\sqrt{6}$
- So $\sqrt{150r} = 5\sqrt{6r}$

Now multiply by $-6$:
$$
-6 \cdot 5\sqrt{6r} = -30\sqrt{6r}
$$

✔ Answer: $-30\sqrt{6r}$

---

20) $5\sqrt{80u^2}$

Simplify $\sqrt{80u^2}$

- $80 = 16 \cdot 5 = 4^2 \cdot 5$, so $\sqrt{80} = 4\sqrt{5}$
- $\sqrt{u^2} = u$

So:
$$
\sqrt{80u^2} = 4u\sqrt{5}
$$

Multiply by 5:
$$
5 \cdot 4u\sqrt{5} = 20u\sqrt{5}
$$

✔ Answer: $20u\sqrt{5}$

---

21) $2\sqrt{125v}$

Simplify $\sqrt{125v}$

- $125 = 25 \cdot 5 = 5^2 \cdot 5$, so $\sqrt{125} = 5\sqrt{5}$
- So $\sqrt{125v} = 5\sqrt{5v}$

Multiply by 2:
$$
2 \cdot 5\sqrt{5v} = 10\sqrt{5v}
$$

✔ Answer: $10\sqrt{5v}$

---

22) $-8\sqrt{24k^2}$

Simplify $\sqrt{24k^2}$

- $24 = 4 \cdot 6 = 2^2 \cdot 6$, so $\sqrt{24} = 2\sqrt{6}$
- $\sqrt{k^2} = k$

So:
$$
\sqrt{24k^2} = 2k\sqrt{6}
$$

Multiply by $-8$:
$$
-8 \cdot 2k\sqrt{6} = -16k\sqrt{6}
$$

✔ Answer: $-16k\sqrt{6}$

---

23) $-4\sqrt{192x}$

Simplify $\sqrt{192x}$

- $192 = 64 \cdot 3 = 8^2 \cdot 3$, so $\sqrt{192} = 8\sqrt{3}$
- So $\sqrt{192x} = 8\sqrt{3x}$

Multiply by $-4$:
$$
-4 \cdot 8\sqrt{3x} = -32\sqrt{3x}
$$

✔ Answer: $-32\sqrt{3x}$

---

24) $2\sqrt{8p^2q^4r}$

Simplify $\sqrt{8p^2q^4r}$

- $8 = 4 \cdot 2 = 2^2 \cdot 2$, so $\sqrt{8} = 2\sqrt{2}$
- $\sqrt{p^2} = p$
- $\sqrt{q^4} = q^2$
- $\sqrt{r}$ stays as is

So:
$$
\sqrt{8p^2q^4r} = 2\sqrt{2} \cdot p \cdot q^2 \cdot \sqrt{r} = 2pq^2\sqrt{2r}
$$

Now multiply by 2:
$$
2 \cdot 2pq^2\sqrt{2r} = 4pq^2\sqrt{2r}
$$

✔ Answer: $4pq^2\sqrt{2r}$

---

25) $-4\sqrt{216x^2y^2z}$

Simplify $\sqrt{216x^2y^2z}$

- $216 = 36 \cdot 6 = 6^2 \cdot 6$, so $\sqrt{216} = 6\sqrt{6}$
- $\sqrt{x^2} = x$
- $\sqrt{y^2} = y$
- $\sqrt{z}$ stays

So:
$$
\sqrt{216x^2y^2z} = 6xy\sqrt{6z}
$$

Multiply by $-4$:
$$
-4 \cdot 6xy\sqrt{6z} = -24xy\sqrt{6z}
$$

✔ Answer: $-24xy\sqrt{6z}$

---

26) $-3\sqrt{24u^4b^2c^2}$

Simplify $\sqrt{24u^4b^2c^2}$

- $24 = 4 \cdot 6 = 2^2 \cdot 6$, so $\sqrt{24} = 2\sqrt{6}$
- $\sqrt{u^4} = u^2$
- $\sqrt{b^2} = b$
- $\sqrt{c^2} = c$

So:
$$
\sqrt{24u^4b^2c^2} = 2u^2bc\sqrt{6}
$$

Multiply by $-3$:
$$
-3 \cdot 2u^2bc\sqrt{6} = -6u^2bc\sqrt{6}
$$

✔ Answer: $-6u^2bc\sqrt{6}$

---

27) $3\sqrt{16x^2y^4z}$

Simplify $\sqrt{16x^2y^4z}$

- $\sqrt{16} = 4$
- $\sqrt{x^2} = x$
- $\sqrt{y^4} = y^2$
- $\sqrt{z}$ stays

So:
$$
\sqrt{16x^2y^4z} = 4xy^2\sqrt{z}
$$

Multiply by 3:
$$
3 \cdot 4xy^2\sqrt{z} = 12xy^2\sqrt{z}
$$

✔ Answer: $12xy^2\sqrt{z}$

---

28) $-2\sqrt{48u^3b^4c^2}$

Simplify $\sqrt{48u^3b^4c^2}$

- $48 = 16 \cdot 3 = 4^2 \cdot 3$, so $\sqrt{48} = 4\sqrt{3}$
- $u^3 = u^2 \cdot u$, so $\sqrt{u^3} = u\sqrt{u}$
- $\sqrt{b^4} = b^2$
- $\sqrt{c^2} = c$

So:
$$
\sqrt{48u^3b^4c^2} = 4\sqrt{3} \cdot u\sqrt{u} \cdot b^2 \cdot c = 4b^2cu\sqrt{3u}
$$

Multiply by $-2$:
$$
-2 \cdot 4b^2cu\sqrt{3u} = -8b^2cu\sqrt{3u}
$$

✔ Answer: $-8b^2cu\sqrt{3u}$

---

29) $6\sqrt{75mp^2q^3}$

Simplify $\sqrt{75mp^2q^3}$

- $75 = 25 \cdot 3 = 5^2 \cdot 3$, so $\sqrt{75} = 5\sqrt{3}$
- $\sqrt{p^2} = p$
- $q^3 = q^2 \cdot q$, so $\sqrt{q^3} = q\sqrt{q}$
- $\sqrt{m}$ stays

So:
$$
\sqrt{75mp^2q^3} = 5\sqrt{3} \cdot p \cdot q\sqrt{q} \cdot \sqrt{m} = 5pq\sqrt{3mq}
$$

Multiply by 6:
$$
6 \cdot 5pq\sqrt{3mq} = 30pq\sqrt{3mq}
$$

✔ Answer: $30pq\sqrt{3mq}$

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30) $4\sqrt{36x^2y^2z^4}$

Simplify $\sqrt{36x^2y^2z^4}$

- $\sqrt{36} = 6$
- $\sqrt{x^2} = x$
- $\sqrt{y^2} = y$
- $\sqrt{z^4} = z^2$

So:
$$
\sqrt{36x^2y^2z^4} = 6xyz^2
$$

Multiply by 4:
$$
4 \cdot 6xyz^2 = 24xyz^2
$$

✔ Answer: $24xyz^2$

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## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 15) | $6xy\sqrt{y}$ |
| 16) | $8x^2y\sqrt{6xy}$ |
| 17) | $28m\sqrt{6m}$ |
| 18) | $36x\sqrt{2}$ |
| 19) | $-30\sqrt{6r}$ |
| 20) | $20u\sqrt{5}$ |
| 21) | $10\sqrt{5v}$ |
| 22) | $-16k\sqrt{6}$ |
| 23) | $-32\sqrt{3x}$ |
| 24) | $4pq^2\sqrt{2r}$ |
| 25) | $-24xy\sqrt{6z}$ |
| 26) | $-6u^2bc\sqrt{6}$ |
| 27) | $12xy^2\sqrt{z}$ |
| 28) | $-8b^2cu\sqrt{3u}$ |
| 29) | $30pq\sqrt{3mq}$ |
| 30) | $24xyz^2$ |

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