Simplifying Radicals of Index 2 (Assume all variables are positive.) - Free Printable

Final Answer:
1) $5\sqrt{5}$
2) $6\sqrt{2}$
3) $3\sqrt{7}$
4) $3\sqrt{5}$
5) $2\sqrt{3}$
6) $3\sqrt{2}$
7) $4\sqrt{6}$
8) $5\sqrt{6}$
9) $2\sqrt{28}$ → Wait, that’s not simplified. Let's correct: $\sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7}$
10) $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$
11) $\sqrt{8a^7} = \sqrt{4a^6 \cdot 2a} = 2a^3\sqrt{2a}$
12) $\sqrt{200b^6} = \sqrt{100b^6 \cdot 2} = 10b^3\sqrt{2}$
13) $\sqrt{20x^5} = \sqrt{4x^4 \cdot 5x} = 2x^2\sqrt{5x}$
14) $\sqrt{45n} = \sqrt{9 \cdot 5n} = 3\sqrt{5n}$
15) $\sqrt{98x^5} = \sqrt{49x^4 \cdot 2x} = 7x^2\sqrt{2x}$
16) $\sqrt{24a^4} = \sqrt{4a^4 \cdot 6} = 2a^2\sqrt{6}$
17) $\sqrt{75x^2} = \sqrt{25x^2 \cdot 3} = 5x\sqrt{3}$
18) $\sqrt{32n^3} = \sqrt{16n^2 \cdot 2n} = 4n\sqrt{2n}$
19) $\sqrt{98a^{12}} = \sqrt{49a^{12} \cdot 2} = 7a^6\sqrt{2}$
20) $\sqrt{28k^9} = \sqrt{4k^8 \cdot 7k} = 2k^4\sqrt{7k}$
21) $\sqrt{18t^5y} = \sqrt{9t^4 \cdot 2ty} = 3t^2\sqrt{2ty}$
22) $\sqrt{24am^5} = \sqrt{4m^4 \cdot 6am} = 2m^2\sqrt{6am}$
23) $\sqrt{50x^5y^4} = \sqrt{25x^4y^4 \cdot 2x} = 5x^2y^2\sqrt{2x}$
24) $\sqrt{320mn^4} = \sqrt{64n^4 \cdot 5m} = 8n^2\sqrt{5m}$
25) $\sqrt{192a^3b} = \sqrt{64a^2 \cdot 3ab} = 8a\sqrt{3ab}$
26) $\sqrt{80x^5y^4} = \sqrt{16x^4y^4 \cdot 5x} = 4x^2y^2\sqrt{5x}$
27) $\sqrt{112x^4y^3} = \sqrt{16x^4y^2 \cdot 7y} = 4x^2y\sqrt{7y}$
28) $\sqrt{256x^3y^5} = \sqrt{256x^2y^4 \cdot xy} = 16xy^2\sqrt{xy}$
29) $\sqrt{216x^3y^2} = \sqrt{36x^2y^2 \cdot 6x} = 6xy\sqrt{6x}$
30) $\sqrt{192a^4c^5} = \sqrt{64a^4c^4 \cdot 3c} = 8a^2c^2\sqrt{3c}$
31) $\sqrt{\frac{252x^4y}{z^2}} = \frac{\sqrt{36x^4 \cdot 7y}}{z} = \frac{6x^2\sqrt{7y}}{z}$
32) $\sqrt{\frac{108m}{n^2p^2}} = \frac{\sqrt{36 \cdot 3m}}{np} = \frac{6\sqrt{3m}}{np}$
33) $\sqrt{\frac{150n}{m^2p^2}} = \frac{\sqrt{25 \cdot 6n}}{mp} = \frac{5\sqrt{6n}}{mp}$
34) $\sqrt{\frac{z^3}{100x^2y^3}} = \frac{\sqrt{z^2 \cdot z}}{10xy\sqrt{y}} = \frac{z\sqrt{zy}}{10xy^2}$ — better to rationalize:
Actually, simplify inside first:
$\sqrt{\frac{z^3}{100x^2y^3}} = \frac{\sqrt{z^2 \cdot z}}{\sqrt{100x^2y^2 \cdot y}} = \frac{z\sqrt{z}}{10xy\sqrt{y}} = \frac{z\sqrt{zy}}{10xy^2}$
Yes, that’s correct.
35) $\sqrt{\frac{72x^3}{x^5y^2}} = \sqrt{\frac{72}{x^2y^2}} = \frac{\sqrt{36 \cdot 2}}{xy} = \frac{6\sqrt{2}}{xy}$
36) $\sqrt{\frac{144pr}{25q^4}} = \frac{12\sqrt{pr}}{5q^2}$
37) $\sqrt{\frac{20mq^2}{81p^2}} = \frac{\sqrt{4 \cdot 5mq^2}}{9p} = \frac{2q\sqrt{5m}}{9p}$
38) $\sqrt{\frac{180}{49x^2y^2z^2}} = \frac{\sqrt{36 \cdot 5}}{7xyz} = \frac{6\sqrt{5}}{7xyz}$
39) $\sqrt{\frac{xy^2z}{225}} = \frac{y\sqrt{xz}}{15}$
40) $\sqrt{\frac{63xyz^2}{16x^4}} = \sqrt{\frac{63yz^2}{16x^3}} = \frac{z\sqrt{63y}}{4x\sqrt{x}} = \frac{z\sqrt{63yx}}{4x^2} = \frac{z\cdot 3\sqrt{7xy}}{4x^2} = \frac{3z\sqrt{7xy}}{4x^2}$

Let me double-check a few key ones for errors:

- #9: $\sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7}$ ✔
- #28: $256 = 16^2$, $x^3 = x^2 \cdot x$, $y^5 = y^4 \cdot y$ → $\sqrt{256x^2y^4} = 16xy^2$, left with $\sqrt{xy}$ → $16xy^2\sqrt{xy}$ ✔
- #35: $\frac{72x^3}{x^5y^2} = \frac{72}{x^2y^2}$, $\sqrt{72} = 6\sqrt{2}$, so $\frac{6\sqrt{2}}{xy}$ ✔
- #40: $\frac{63xyz^2}{16x^4} = \frac{63yz^2}{16x^3}$. Pull out squares: $z^2$ → $z$, $x^3 = x^2 \cdot x$, so denominator $4x\sqrt{x}$, numerator $z\sqrt{63y} = z\cdot 3\sqrt{7y}$, combine: $\frac{3z\sqrt{7y}}{4x\sqrt{x}} = \frac{3z\sqrt{7xy}}{4x^2}$ ✔

All look correct.

But the instruction says: *Only then provide the final answer*, and the format expects only the final answer in the Final Answer section — but there are 40 items. Since the user didn’t specify to list all, but the task is “Simplify radicals”, and the image has 40 problems, the expected output is the simplified forms for all 40.

However, the response format says:
Final Answer: [Provide only the final correct answer] — implying one box, but 40 answers.

Given standard practice for such worksheets, we list all simplified results clearly.

I will now write the final answer as a numbered list of simplified radicals.

Final Answer:
1) $5\sqrt{5}$
2) $6\sqrt{2}$
3) $3\sqrt{7}$
4) $3\sqrt{5}$
5) $2\sqrt{3}$
6) $3\sqrt{2}$
7) $4\sqrt{6}$
8) $5\sqrt{6}$
9) $4\sqrt{7}$
10) $5\sqrt{2}$
11) $2a^3\sqrt{2a}$
12) $10b^3\sqrt{2}$
13) $2x^2\sqrt{5x}$
14) $3\sqrt{5n}$
15) $7x^2\sqrt{2x}$
16) $2a^2\sqrt{6}$
17) $5x\sqrt{3}$
18) $4n\sqrt{2n}$
19) $7a^6\sqrt{2}$
20) $2k^4\sqrt{7k}$
21) $3t^2\sqrt{2ty}$
22) $2m^2\sqrt{6am}$
23) $5x^2y^2\sqrt{2x}$
24) $8n^2\sqrt{5m}$
25) $8a\sqrt{3ab}$
26) $4x^2y^2\sqrt{5x}$
27) $4x^2y\sqrt{7y}$
28) $16xy^2\sqrt{xy}$
29) $6xy\sqrt{6x}$
30) $8a^2c^2\sqrt{3c}$
31) $\dfrac{6x^2\sqrt{7y}}{z}$
32) $\dfrac{6\sqrt{3m}}{np}$
33) $\dfrac{5\sqrt{6n}}{mp}$
34) $\dfrac{z\sqrt{zy}}{10xy^2}$
35) $\dfrac{6\sqrt{2}}{xy}$
36) $\dfrac{12\sqrt{pr}}{5q^2}$
37) $\dfrac{2q\sqrt{5m}}{9p}$
38) $\dfrac{6\sqrt{5}}{7xyz}$
39) $\dfrac{y\sqrt{xz}}{15}$
40) $\dfrac{3z\sqrt{7xy}}{4x^2}$