# 10.9: Radicals- Answers to the Homework Exercises

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1. $$7\sqrt{5}$$
1. $$2\sqrt{3}$$
1. $$48\sqrt{2}$$
1. $$8\sqrt{3n}$$
1. $$6x\sqrt{7}$$
1. $$-56x^2$$
1. $$3xy\sqrt{5}$$
1. $$8x^2y^2\sqrt{5}$$
1. $$35xy\sqrt{5y}$$
1. $$-48x^2z^2y\sqrt{5}$$
1. $$-12p\sqrt{6mn}$$
1. $$14$$
1. $$20\sqrt{2}$$
1. $$-21\sqrt{7}$$
1. $$10n\sqrt{n}$$
1. $$-20p^2\sqrt{7}$$
1. $$32p\sqrt{7}$$
1. $$16a^2b\sqrt{2}$$
1. $$56\sqrt{2mn}$$
1. $$-30y^2x\sqrt{2x}$$
1. $$-4yz\sqrt{2xz}$$
1. $$(\sqrt[5]{m})^3$$
1. $$(\sqrt{7x})^3$$
1. $$(6x)^{-\dfrac{3}{2}}$$
1. $$n^{-\dfrac{7}{4}}$$
1. $$4$$
1. $$8$$

1. $$6\sqrt{5}$$
1. $$-5\sqrt{6}$$
1. $$-8\sqrt{2}$$
1. $$-2\sqrt{2}$$
1. $$-3\sqrt{6}-\sqrt{3}$$
1. $$-4\sqrt{6}+4\sqrt{5}$$
1. $$2\sqrt[3]{2}$$
1. $$\sqrt[4]{2}-3\sqrt[4]{3}$$
1. $$2\sqrt[4]{2}+\sqrt[4]{3}+6\sqrt[4]{4}$$
1. $$4\sqrt[5]{5}-4\sqrt[5]{6}$$
1. $$-3\sqrt{6}-5\sqrt{3}$$
1. $$-3\sqrt{3}$$
1. $$-6\sqrt{6}+9\sqrt{3}$$
1. $$8\sqrt{5}-\sqrt{3}$$
1. $$3\sqrt{2}+3\sqrt{6}$$
1. $$-\sqrt{5}-3\sqrt{6}$$
1. $$6\sqrt[3]{5}-3\sqrt[3]{3}$$
1. $$5\sqrt[4]{6}+2\sqrt[4]{4}$$
1. $$-2\sqrt[4]{3}-9\sqrt[4]{5}-3\sqrt[4]{2}$$
1. $$-11\sqrt[7]{2}-2\sqrt[7]{5}$$

## Multiply and Divide Radicals

1. $$-48\sqrt{5}$$
1. $$2x^2\sqrt[3]{x}$$
1. $$-45\sqrt{5}-10\sqrt{15}$$
1. $$-2-4\sqrt{2}$$
1. $$6a+a\sqrt{10}+6a\sqrt{6}+2a\sqrt{15}$$
1. $$\dfrac{\sqrt{3}}{25}$$
1. $$\dfrac{\sqrt{5}}{2}$$
1. $$\dfrac{5}{12y^4}$$
1. $$\dfrac{\sqrt[3]{10}}{5}$$
1. $$\dfrac{5\sqrt[4]{r^2}}{2}$$ or $$\dfrac{5\sqrt{r}}{2}$$
1. $$-25r^2\sqrt{2r}$$
1. $$5\sqrt{2}+2\sqrt{5}$$
1. $$5\sqrt{3}-9\sqrt{5v}$$
1. $$30+8\sqrt{3}+5\sqrt{15}+4\sqrt{5}$$
1. $$-10\sqrt{m}+25\sqrt{2}+\sqrt{2m}-5$$
1. $$2$$
1. $$4\sqrt{2}$$
1. $$\dfrac{\sqrt{n}}{2}$$
1. $$\dfrac{1}{4}$$

## Rationalize Denominators

1. $$\dfrac{4\sqrt{3}}{9}$$
1. $$2$$
1. $$\dfrac{4\sqrt{5}}{5}$$
1. $$\dfrac{2+\sqrt{3}}{5}$$
1. $$\dfrac{\sqrt{6}-9}{3}$$
1. $$\dfrac{10-2\sqrt{2}}{23}$$
1. $$3-\sqrt{5}$$
1. $$\sqrt{2}-1$$
1. $$\sqrt{a}$$
1. $$4-2\sqrt{3}+2\sqrt{6}-3\sqrt{2}$$
1. $$3\sqrt{2}+2\sqrt{3}$$
1. $$\dfrac{-1+\sqrt{5}}{4}$$
1. $$\dfrac{\sqrt{3}-1}{4}$$
1. $$\dfrac{\sqrt{30}-2\sqrt{3}}{18}$$
1. $$\dfrac{2\sqrt{3}+\sqrt{2}}{2}$$
1. $$\dfrac{\sqrt{5}-\sqrt{3}}{2}$$
1. $$3+2\sqrt{3}$$
1. $$3-2\sqrt{2}$$
1. $$\dfrac{2\sqrt{5}-2\sqrt{15}+\sqrt{3}+3}{-2}$$
1. $$\dfrac{a\sqrt{b}+b\sqrt{a}}{a-b}$$
1. $$\dfrac{2\sqrt{5}-5\sqrt{2}-10+5\sqrt{10}}{30}$$
1. $$\dfrac{8+3\sqrt{6}}{10}$$

## Radicals with Mixed Indices

1. $$\sqrt[4]{4x^2y^3}$$
1. $$\dfrac{\sqrt[3]{36xy}}{3y}$$
1. $$\sqrt[4]{x^3y^2z}$$
1. $$\sqrt{3xy^3}$$
1. $$\sqrt[5]{x^3y^4z^2}$$
1. $$\sqrt{5y}$$
1. $$\sqrt[6]{5400}$$
1. $$\sqrt[6]{x^3(x-2)^2}$$
1. $$\sqrt[12]{x^{11}y^{10}}$$
1. $$a\sqrt[4]{a}$$
1. $$xy\sqrt[6]{xy^5}$$
1. $$x\sqrt[12]{59049xy^{11}z^{10}}$$
1. $$\sqrt[12]{a^5}$$
1. $$\sqrt[10]{ab^9c^7}$$
1. $$\sqrt[15]{(2x+1)^4}$$
1. $$\sqrt[15]{27y^5z^5}$$
1. $$\sqrt[10]{4a^9b^9}$$
1. $$\sqrt[30]{x^{22}y^{11}z^{27}}$$
1. $$a\sqrt[12]{a^5}$$
1. $$2xy^2\sqrt[6]{2x^5y}$$
1. $$4x(y+z)^3\sqrt[6]{2x(y+z)}$$
1. $$\dfrac{\sqrt[15]{a^7b^{11}}}{b}$$
1. $$\sqrt[12]{(2+5x)^5}$$

1. $$3$$
1. $$\pm 2$$
1. $$5$$
1. $$5$$
1. $$3$$
1. $$3$$
1. $$7$$
1. $$21$$
1. $$79.5$$ inches
2. $$145.7$$ pounds

## Solving with Rational Exponents

1. $$\pm 5\sqrt{3}$$
1. $$\pm 2\sqrt{6}$$
1. $$-1$$
1. $$-7$$
1. $$-\dfrac{3}{8}, -\dfrac{5}{8}$$
1. $$-\dfrac{34}{3}, -10$$
1. $$-2$$
1. $$-3,11$$
1. $$\dfrac{-1\pm 3\sqrt{2}}{2}$$
1. $$-\dfrac{11}{2},\dfrac{5}{2}$$
1. $$\dfrac{9}{8}$$
1. $$3$$

## Complex Numbers

1. $$8i$$
1. $$9i$$
1. $$10i$$
1. $$3i\sqrt{10}$$
1. $$7i\sqrt{5}$$
1. $$40$$
1. $$-12$$
1. $$-15$$
1. $$-2\sqrt{5}$$
1. $$-7-4i$$
1. $$2-5i$$
1. $$\dfrac{1+i\sqrt{3}}{2}$$
1. $$\dfrac{2+i\sqrt{2}}{2}$$
1. $$5-i\sqrt{3}$$
1. $$\dfrac{5+2i\sqrt{3}}{2}$$
1. $$11-4i$$
1. $$-3-13i$$
1. $$-8-2i$$
1. $$80-10i$$
1. $$44+8i$$
1. $$9i+5$$
1. $$\dfrac{3i-6}{4}$$
1. $$\dfrac{-40i+4}{101}$$
1. $$\dfrac{70+49i}{149}$$
1. $$-4i$$
1. $$5-12i$$
1. $$13-8i$$
1. $$-32-128i$$
1. $$-28+76i$$
1. $$-1+13i$$
1. $$\dfrac{4i+2}{3}$$
1. $$-2i$$
1. $$\dfrac{4-6i}{13}$$
1. $$\dfrac{48i-56}{85}$$
1. $$1$$
1. $$-1$$
1. $$1$$
1. $$-i$$

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