Chapter 13

Addition and Subtraction of Radicals. Combine as indicated and simplify. $$3 \sqrt{\frac{1}{3}}-2 \sqrt{\frac{3}{4}}+4 \sqrt{3}$$

Simplify, and write without negative exponents. Do some by calculator. $$\left(a^{p}-b^{q}\right)^{2}$$

Solve for \(x\) and check. $$\frac{6}{\sqrt{3+x}}=\sqrt{x+3}$$

Express in exponential form. $$\sqrt{x^{2} y^{2}}$$

Addition and Subtraction of Radicals. Combine as indicated and simplify. $$7 \sqrt{\frac{27}{50}}-3 \sqrt{\frac{2}{3}}$$

Simplify, and write without negative exponents. Do some by calculator. $$(x+y)^{-1}$$

Solve for \(x\) and check. $$\sqrt{12+x}=2+\sqrt{x}$$

Express in exponential form. $$\sqrt[n]{a^{n} b^{3 n}}$$

Simplify, and write without negative exponents. Do some by calculator. $$(2 a+3 b)^{-1}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{18}$$

Find the missing side and the hypotenuse of a right triangle that has one side equal to 293 in. and a perimeter of 994 in.

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{75}$$

Multiplication of Radicals. Multiply and simplify. $$3 \sqrt{3} \mathrm{by} 5 \sqrt{3}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{63}$$

Multiplication of Radicals. Multiply and simplify. $$2 \sqrt{3} \text { by } 3 \sqrt{8}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt[3]{16}$$

Simplify, and write without negative exponents. Do some by calculator. $$p^{-1}-3 q^{-2}$$

Multiplication of Radicals. Multiply and simplify. $$\sqrt{8} \text { by } \sqrt{160}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt[3]{56}$$

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{3}{b}\right)^{-1}$$

Multiplication of Radicals. Multiply and simplify. $$\sqrt{\frac{5}{8}} \mathrm{by} \sqrt{\frac{3}{4}}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt[4]{48}$$

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{x}{4}\right)^{-1}$$

Multiplication of Radicals. Multiply and simplify. $$4 \sqrt[3]{45} \text{ by } 2 \sqrt[3]{3}$$

Writing: Explain in your own words how to solve a radical equation graphically. Be sure to give the reasons behind this method. Also say if the method can or cannot be used for other types of equations, and why.

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{2 x}{3 y}\right)^{-2}$$

Multiplication of Radicals. Multiply and simplify. $$3 \sqrt{3} \text { by } 2 \sqrt[3]{2}$$

Project: Sometimes when solving a radical equation, the squaring operation will result in a quadratic equation, which can then be solved by the methods of the preceding chapter. Try solving these equations, and be sure to check all solutions. (a) \(3 \sqrt{x-1}-\frac{4}{\sqrt{x-1}}=4\) (b) \(\sqrt{5 x^{2}-3 x-41}=3 x-7\) (c) \(\sqrt{x+5}=\frac{12}{\sqrt{x+12}}\) (d) \(\sqrt{7 x+8}-\sqrt{5 x-4}=2\)

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$3 \sqrt{50 x^{5}}$$

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{3 p}{4 y}\right)^{-1}$$

Multiplication of Radicals. Multiply and simplify. $$3 \sqrt{2} \text { by } 2 \sqrt[3]{3}$$

Project, Golden Ratio: The value of the golden ratio \(\Phi\) is given by \(\Phi=\frac{1+\sqrt{5}}{2}\) It is also given by the following radical equation. Demonstrate by calculation, by hand, or with a spreadsheet that this is true. $$ \Phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}} $$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{36 x^{2} y}$$

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{2 p x}{3 q y}\right)^{-3}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt[3]{x^{2} y^{5}}$$

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{p^{2}}{q^{0}}\right)^{-1}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{\frac{3}{7}}$$

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{3 a^{4} b^{3}}{5 x^{2} y}\right)^{2}$$

Multiplication of Radicals. Multiply and simplify. $$2 \sqrt[3]{24} \text{ by } \sqrt[9]{\frac{8}{27}}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{\frac{2}{3}}$$

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{x^{2}}{y}\right)^{-3}$$

Multiplication of Radicals. Multiply and simplify. $$2 x \sqrt{3 a} \text { by } 3 \sqrt{y}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt[3]{\frac{1}{4}}$$

Challenge Problems. $$(3 m)^{-3}-2 n^{-2}$$

Multiplication of Radicals. Multiply and simplify. $$3 \sqrt[3]{9 a^{2}} \text { by } \sqrt[3]{3 a b c}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{\frac{5}{8}}$$

Challenge Problems. $$(5 a)^{-2}+\left(2 a^{2} b^{-4}\right)^{-2}$$

Multiplication of Radicals. Multiply and simplify. $$\sqrt[5]{4 x y^{2}} \text{ by } \sqrt[5]{8 x^{2} y}$$

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt[3]{\frac{2}{9}}$$

Challenge Problems. $$\left(x^{n}+y^{m}\right)^{2}$$