Answer Key 9.4

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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. \(\phantom{1}\)
    \(\dfrac{\sqrt{12}}{5\sqrt{100}}\div \sqrt{4} \\ \)
    \(\dfrac{\sqrt{3}}{5\sqrt{25}}\Rightarrow \dfrac{\sqrt{3}}{5\cdot 5}\Rightarrow \dfrac{\sqrt{3}}{25}\)
22. \(\dfrac{\sqrt{15}}{2\cdot 2}\Rightarrow \dfrac{\sqrt{15}}{4}\)
23. \(\phantom{1}\)
    \(\dfrac{\sqrt{5}}{4\sqrt{125}}\div \sqrt{5} \\ \)
    \(\dfrac{\sqrt{1}}{4\sqrt{25}}\Rightarrow \dfrac{1}{4\cdot 5}\Rightarrow \dfrac{1}{20}\)
24. \(\phantom{1}\)
    \(\dfrac{\sqrt{12}}{\sqrt{3}}\div \sqrt{3} \\ \)
    \(\dfrac{\sqrt{4}}{\sqrt{1}}\Rightarrow \dfrac{2}{1}\Rightarrow 2\)
25. \(\phantom{1}\)
    \(\dfrac{\sqrt{10}}{\sqrt{6}}\div \sqrt{2} \\ \)
    \(\dfrac{\sqrt{5}}{\sqrt{3}}\)
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. \(\phantom{1}\)
    \(\dfrac{\sqrt{2p^2}}{\sqrt{3p}}\div \sqrt{p} \\ \)
    \(\dfrac{\sqrt{2p}}{\sqrt{3}}\)
30. \(\phantom{1}\)
    \(\dfrac{\sqrt{8n^2}}{\sqrt{10n}}\div \sqrt{2n} \\ \)
    \(\dfrac{\sqrt{4n}}{\sqrt{5}}\Rightarrow \dfrac{2\sqrt{n}}{\sqrt{5}}\)