Chapter 5

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{2}{\sqrt{x}+4}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(2 \sqrt{18 x}-3 \sqrt{8 x}-6 \sqrt{50 x}\)

Change each radical to simplest radical form. \(2 \sqrt[3]{81}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\frac{28 x^{-2} y^{-3}}{4 x^{-3} y^{-1}}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(9 x^{2} y^{4}\right)^{\frac{1}{2}}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{3}{\sqrt{x}+7}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(4 \sqrt{20 x}+5 \sqrt{45 x}-10 \sqrt{80 x}\)

Change each radical to simplest radical form. \(-3 \sqrt[3]{54}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\frac{-72 a^{2} b^{-4}}{6 a^{3} b^{-7}}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\frac{24 x^{\frac{3}{5}}}{6 x^{\frac{1}{3}}}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{x}}{\sqrt{x}-5}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(5 \sqrt{27 n}-\sqrt{12 n}-6 \sqrt{3 n}\)

Change each radical to simplest radical form. \(\frac{2}{\sqrt[3]{9}}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\frac{-72 a^{2} b^{-4}}{6 a^{3} b^{-7}}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\frac{18 x^{\frac{1}{2}}}{9 x^{\frac{1}{3}}}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{x}}{\sqrt{x}-1}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(4 \sqrt{8 n}+3 \sqrt{18 n}-2 \sqrt{72 n}\)

Change each radical to simplest radical form. \(\frac{3}{\sqrt[3]{3}}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\frac{108 a^{-5} b^{-4}}{9 a^{-2} b}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\frac{48 b^{\frac{1}{3}}}{12 b^{\frac{3}{4}}}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{x}-2}{\sqrt{x}+6}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(7 \sqrt{4 a b}-\sqrt{16 a b}-10 \sqrt{25 a b}\)

Change each radical to simplest radical form. \(\frac{\sqrt[3]{27}}{\sqrt[3]{4}}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(\frac{35 x^{-1} y^{-2}}{7 x^{4} y^{3}}\right)^{-1}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\frac{56 a^{\frac{1}{6}}}{8 a^{\frac{1}{4}}}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{x}+1}{\sqrt{x}-10}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(4 \sqrt{a b}-9 \sqrt{36 a b}+6 \sqrt{49 a b}\)

Change each radical to simplest radical form. \(\frac{\sqrt[3]{8}}{\sqrt[3]{16}}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(\frac{-48 a b^{2}}{-6 a^{3} b^{5}}\right)^{-2}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{6 x^{\frac{2}{5}}}{7 y^{\frac{2}{3}}}\right)^{2}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{x}}{\sqrt{x}+2 \sqrt{y}}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(-3 \sqrt{2 x^{3}}+4 \sqrt{8 x^{3}}-3 \sqrt{32 x^{3}}\)

Change each radical to simplest radical form. \(\frac{\sqrt[3]{6}}{\sqrt[3]{4}}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(\frac{-36 a^{-1} b^{-6}}{4 a^{-1} b^{4}}\right)^{-2}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{2 x^{\frac{1}{3}}}{3 y^{\frac{1}{4}}}\right)^{4}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{y}}{2 \sqrt{x}-\sqrt{y}}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(2 \sqrt{40 x^{5}}-3 \sqrt{90 x^{5}}+5 \sqrt{160 x^{5}}\)

Change each radical to simplest radical form. \(\frac{\sqrt[3]{4}}{\sqrt[3]{2}}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(\frac{8 x y^{3}}{-4 x^{4} y}\right)^{-3}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{x^{2}}{y^{3}}\right)^{-\frac{1}{2}}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{3 \sqrt{y}}{2 \sqrt{x}-3 \sqrt{y}}\)

Is the expression \(3 \sqrt{2}+\sqrt{50}\) in simplest radical form? Defend your answer.

For Problems \(75-84\), express each of the following as a single fraction involving positive exponents only. \(x^{-2}+x^{-3}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{a^{3}}{b^{-2}}\right)^{-\frac{1}{3}}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{2 \sqrt{x}}{3 \sqrt{x}+5 \sqrt{y}}\)

Your friend simplified \(\frac{\sqrt{6}}{\sqrt{8}}\) as follows: $$ \frac{\sqrt{6}}{\sqrt{8}} \cdot \frac{\sqrt{8}}{\sqrt{8}}=\frac{\sqrt{48}}{8}=\frac{\sqrt{16} \sqrt{3}}{8}=\frac{4 \sqrt{3}}{8}=\frac{\sqrt{3}}{2} $$

For Problems \(75-84\), express each of the following as a single fraction involving positive exponents only. \(x^{-1}+x^{-5}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{18 x^{\frac{1}{3}}}{9 x^{\frac{1}{4}}}\right)^{2}\)

How would you help someone rationalize the denominator and simplify \(\frac{4}{\sqrt{8}+\sqrt{12}}\) ?

Does \(\sqrt{x+y}\) equal \(\sqrt{x}+\sqrt{y}\) ? Defend your answer.