Chapter 3

In \(11-38,\) evaluate each expression in the set of real numbers. $$ -\sqrt[4]{\frac{b^{8}}{1,296}}, b \geq 0 $$

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{63 a^{2}}-\sqrt{45 a^{2}} $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{2}{\sqrt{x}}+\frac{2}{\sqrt{y}}\)

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{300 c} $$

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (\sqrt{6}+6 c)(\sqrt{6}-6 c) $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 1+\sqrt{2 x}=\sqrt{3 x+1} $$

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[5]{-0.00001 y^{5}}, y \geq 0 $$

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{4 a b^{2}}-\sqrt{a b^{2}} $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{1}{\sqrt{x}+6}-\frac{2}{\sqrt{6}}\)

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[3]{\frac{a^{3}}{3}} $$

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (a+\sqrt{b})(a-\sqrt{b}) $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{x+4}-\sqrt{x-1}=1 $$

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[k]{1}, k \text { an integer greater than } 2 $$

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{50 x^{3}}+\sqrt{200 x^{3}} $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{2 \sqrt{a}}{\sqrt{b}}+\frac{2 \sqrt{b}}{\sqrt{a}}\)

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[4]{\frac{2 a^{8}}{b^{2} c^{3}}} $$

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (1-\sqrt{3})^{2} $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{5 x}-\sqrt{x+4}=3 $$

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[k]{x^{2 k}}, k \text { an integer greater than } 2 $$

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{49 x^{3}}-2 x \sqrt{4 x} $$

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{3}{x+\sqrt{2}}+\frac{5}{\sqrt{x}}\)

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[4]{32 x^{5} y^{4}} $$

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \left(3+\sqrt{5 a b^{3}}\right)^{2} $$

The lengths of the legs of an isosceles right triangle are \(x\) and \(\sqrt{6 x+16} .\) What are the lengths of the legs of the triangle?

In \(39-42,\) find the set of real numbers for which the given radical is a real number. $$ \sqrt{x-2} $$

In \(39-42,\) solve and check each equation. $$ 5 x-\sqrt{3}=\sqrt{48} $$

a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form. \(\frac{4}{\sqrt{6}}\)

The lengths of the legs of a right triangle are 8 centimeters and 12 centimeters. Express the length of the hypotenuse in simplest radical form.

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (1-\sqrt{7})(1+\sqrt{7})(1+\sqrt{7}) $$

The width of a rectangle is \(x\) and the length is \(\sqrt{x-1}\) . If the width is twice the length, what are the dimensions of the rectangle?

In \(39-42,\) find the set of real numbers for which the given radical is a real number. $$ \sqrt{9-3 x} $$

In \(39-42,\) solve and check each equation. $$ 12 y+\sqrt{32}=\sqrt{200} $$

a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form. \(\frac{2+\sqrt{3}}{\sqrt{3}}\)

The length of one leg of an isosceles right triangle is 6 inches. Express the length of the hypotenuse in simplest radical form.

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (2-\sqrt{5})(2+\sqrt{5})^{2} $$

In \(\triangle A B C, A B=\sqrt{7 x+5, B C}=\sqrt{5 x+15},\) and \(A C=\sqrt{2 x}\) a. If \(A B=B C,\) find the length of each side of the triangle. b. Express the perimeter of the triangle in simplest radical form.

In \(39-42,\) find the set of real numbers for which the given radical is a real number. $$ \sqrt{4 x+12} $$

In \(39-42,\) solve and check each equation. $$ 4 a+\sqrt{6}=a+\sqrt{96} $$

a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form. \(\frac{4}{\sqrt{3}-1}\)

The length of the hypotenuse of a right triangle is 24 meters and the length of one leg is 12 meters. Express the length of the other leg in simplest radical form.

The length of a side of a square is 48\(\sqrt{2}\) meters. Express the area of the square in simplest form.

In \(39-42,\) find the set of real numbers for which the given radical is a real number. $$ \sqrt[4]{x+5} $$

In \(39-42,\) solve and check each equation. $$ y+\sqrt{20}=\sqrt{45}-2 y $$

a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form. \(\frac{3+\sqrt{7}}{3-\sqrt{7}}\)

The dimensions of a rectangle are 15 feet by 10 feet. Express the length of the diagonal in simplest radical form.

The dimensions of a rectangle are 12\(\sqrt{2}\) feet by \(\sqrt{50}\) feet. Express the area of the rectangle in simplest form.

In \(43-46,\) solve each equation for the variable. $$ x^{2}=81 $$

In \(43-47,\) express each answer in simplest radical form. The lengths of the sides of a triangle are \(\sqrt{75}\) inches, \(\sqrt{27}\) inches, and \(\sqrt{108}\) inches. What is the perimeter of the triangle?

Solve and check each equation. \(2 a+\sqrt{50}=\sqrt{98}\)

The area of a square is 150 square feet. Express the length of a side of the square in simplest radical form.