1. $12\sqrt{5\cdot 16}$
$12\cdot 4 \sqrt{5}$
$48\sqrt{5}$
2. $-5\sqrt{10\cdot 15}$
$-5\sqrt{150}$
$-5\sqrt{25\cdot 6}\Rightarrow -25\sqrt{6}$
3. $\sqrt{15\cdot 12\cdot m^2}$
$\sqrt{3\cdot 5\cdot 3\cdot 4\cdot m^2}$
$3\cdot 2m\sqrt{5}$
$6m\sqrt{5}$
4. $-5\sqrt{5r^3\cdot 10r^2}$
$-5\sqrt{25\cdot 2\cdot r^4\cdot r}$
$-25r^2\sqrt{2r}$
5. $\sqrt[3]{8x^7}$
$\sqrt[3]{8\cdot x^6\cdot x}$
$2x^2 \sqrt[3]{x}$
6. $3 \sqrt[3]{40a^7}$
$3 \sqrt[3]{5\cdot 8\cdot a^6\cdot a}$
$3\cdot 2a^2 \sqrt[3]{5a}\Rightarrow 6a^2 \sqrt[3]{5a}$
7. $\sqrt{12}+2\sqrt{6}$
$\sqrt{4\cdot 3}+2\sqrt{6}$
$2\sqrt{3}+2\sqrt{6}$
8. $\sqrt{50}+\sqrt{20}$
$\sqrt{25\cdot 2}+\sqrt{4\cdot 5}$
$5\sqrt{2}+2\sqrt{5}$
9. $-15\sqrt{45}-10\sqrt{15}$
$-15\sqrt{9\cdot 5}-10\sqrt{15}$
$-15\cdot 3\sqrt{5}-10\sqrt{15}$
$-45\sqrt{5}-10\sqrt{15}$
10. $15\sqrt{45}+10\sqrt{15}$
$15\sqrt{9\cdot 5}+10\sqrt{15}$
$15\cdot 3\sqrt{5}+10\sqrt{15}$
$45\sqrt{5}+10\sqrt{15}$
11. $25n\sqrt{10}+5\sqrt{20}$
$25n\sqrt{10}+5\sqrt{4\cdot 5}$
$25n\sqrt{10}+10\sqrt{5}$
12. $\sqrt{75}-3\sqrt{45v}$
$\sqrt{25\cdot 3}-3\sqrt{9\cdot 5v}$
$5\sqrt{3}-9\sqrt{5v}$
13. $-6+2\sqrt{2}-6\sqrt{2}+2(\sqrt{2})(\sqrt{2})$
$-6+2\sqrt{2}-6\sqrt{2}+2(2)$
$-6+4+2\sqrt{2}-6\sqrt{2}$
$-2-4\sqrt{2}$
14. $10-4\sqrt{3}-5\sqrt{3}+2(\sqrt{3})(\sqrt{3})$
$10-4\sqrt{3}-5\sqrt{3}+2(3)$
$10+6-4\sqrt{3}-5\sqrt{3}$
$16-9\sqrt{3}$
15. $(2\sqrt{5})(\sqrt{5})-\sqrt{5}-10\sqrt{5}+5$
$2(5)-\sqrt{5}-10\sqrt{5}+5$
$10+5-\sqrt{5}-10\sqrt{5}$
$15-11\sqrt{5}$
16. $10(3)+4\sqrt{12}+5\sqrt{15}+2\sqrt{20}$
$30+4\sqrt{4\cdot 3}+5\sqrt{15}+2\sqrt{5\cdot 4}$
$30+5\sqrt{15}+8\sqrt{3}+4\sqrt{5}$
17. $3(2a)+6\sqrt{6a^2}+\sqrt{10a^2}+2\sqrt{15a^2}$
$6a+6a\sqrt{6}+a\sqrt{10}+2a\sqrt{15}$
18. $(-2\sqrt{2p}+5\sqrt{5})(2\sqrt{5p})$
$-4\sqrt{10p^2}+10\sqrt{25p}$
$-4p\sqrt{10}+50\sqrt{p}$
19. $15+12\sqrt{3}+20\sqrt{3}+16(3)$
$63+32\sqrt{3}$
20. $-5\sqrt{4m}+\sqrt{2m}+25\sqrt{2}-5$
$-10\sqrt{m}+\sqrt{2m}+25\sqrt{2}-5$
21. $\dfrac{\sqrt{12}}{5\sqrt{100}}\div \sqrt{4}$
$\phantom{a}$
$\dfrac{\sqrt{3}}{5\sqrt{25}}\Rightarrow \dfrac{\sqrt{3}}{5\cdot 5}\Rightarrow \dfrac{\sqrt{3}}{25}$
22. $\dfrac{\sqrt{15}}{2\cdot 2}\Rightarrow \dfrac{\sqrt{15}}{4}$
23. $\dfrac{\sqrt{5}}{4\sqrt{125}}\div \sqrt{5}$
$\phantom{a}$
$\dfrac{\sqrt{1}}{4\sqrt{25}}\Rightarrow \dfrac{1}{4\cdot 5}\Rightarrow \dfrac{1}{20}$
24. $\dfrac{\sqrt{12}}{\sqrt{3}}\div \sqrt{3}$
$\phantom{a}$
$\dfrac{\sqrt{4}}{\sqrt{1}}\Rightarrow \dfrac{2}{1}\Rightarrow 2$
25. $\dfrac{\sqrt{10}}{\sqrt{6}}\div \sqrt{2}$
$\phantom{a}$
$\dfrac{\sqrt{5}}{\sqrt{3}}$
26. Does not reduce
27. $\dfrac{5x^2}{4\sqrt{3\cdot x^2\cdot x\cdot y^2\cdot y}}\Rightarrow \dfrac{5x^2}{4xy\sqrt{3xy}}\Rightarrow \dfrac{5x}{4y\sqrt{3xy}}$
28. $\dfrac{4}{5y^2\sqrt{3x}}$
29. $\dfrac{\sqrt{2p^2}}{\sqrt{3p}}\div \sqrt{p}$
$\phantom{a}$
$\dfrac{\sqrt{2p}}{\sqrt{3}}$
30. $\dfrac{\sqrt{8n^2}}{\sqrt{10n}}\div \sqrt{2n}$
$\phantom{a}$
$\dfrac{\sqrt{4n}}{\sqrt{5}}\Rightarrow \dfrac{2\sqrt{n}}{\sqrt{5}}$