Chapter 7

Factor the given factor from the expression. $$ x^{-1 / 3} ; 5 x^{-1 / 3}+x^{2 / 3} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (5-6 i)-4 i $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{3 \sqrt{100 x^{2}}}{2 \sqrt{2 x^{-1}}} $$

Multiply and then simplify if possible. $$ (\sqrt[3]{x}+1)(\sqrt[3]{x^{2}}-\sqrt[3]{x}+1) $$

Find the length of a pendulum whose period is 4 seconds. Round your answer to 2 decimal places.

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{y^{10}}{9 x^{6}}} $$

Factor the given factor from the expression. $$ x^{-3 / 4} ; x^{-3 / 4}+3 x^{1 / 4} $$

Find the length of a pendulum whose period is 3 seconds. Round your answer to 2 decimal places.

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (6-2 i)+7 i $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt{270 y^{2}}}{5 \sqrt{3 y^{-4}}} $$

Multiply and then simplify if possible. $$ (\sqrt[3]{3 x}+2)(\sqrt[3]{9 x^{2}}-2 \sqrt[3]{3 x}+4) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ -\sqrt[3]{\frac{z^{21}}{27 x^{3}}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{x^{3}} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{2-3 i}{2+i} $$

Multiply and then simplify if possible. $$ (\sqrt{x-1}+5)^{2} $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt[4]{96 a^{10} b^{3}}}{\sqrt[4]{3 a^{2} b^{3}}} $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ -\sqrt[3]{\frac{64 a^{3}}{b^{9}}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[9]{a^{3}} $$

Galileo experimented with pendulums. He supposedly made conjectures about pendulums of equal length with different bob weights. Try this experiment. Make two pendulums 3 feet long. Attach a heavy weight (lead) to one and a light weight (a cork) to the other. Pull both pendulums back the same angle measure and release. Make a conjecture from your observations.

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{6+5 i}{6-5 i} $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt[4]{160 x^{10} y^{5}}}{\sqrt[4]{2 x^{2} y^{2}}} $$

Rationalize each numerator. See Example 7. $$ \frac{\sqrt{15}+1}{2} $$

Multiply and then simplify if possible. $$ (\sqrt{3 x+1}+2)^{2} $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{\frac{x^{4}}{16}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{4} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (2+4 i)+(6-5 i) $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt[5]{64 x^{10} y^{3}}}{\sqrt[5]{2 x^{3} y^{-7}}} $$

Multiply and then simplify if possible. $$ (\sqrt{2 x+5}-1)^{2} $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{\frac{y^{4}}{81 x^{4}}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[4]{36} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (5-3 i)+(7-8 i) $$

Rationalize each numerator. See Example 7. $$ \frac{\sqrt{5}+2}{\sqrt{2}} $$

Multiply and then simplify if possible. $$ (\sqrt{x-6}-7)^{2} $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt[5]{192 x^{6} y^{12}}}{\sqrt[5]{2 x^{-1} y^{-3}}} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ f(0) $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[4]{16 x^{2}} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (\sqrt{6}+i)(\sqrt{6}-i) $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (5,1) \text { and }(8,5) $$

Rationalize each numerator. See Example 7. $$ \frac{\sqrt{x}+3}{\sqrt{x}} $$

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{2 x-14}{2} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ g(0) $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[8]{4 y^{2}} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (\sqrt{6}+i)(\sqrt{6}-i) $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (2,3) \text { and }(14,8) $$

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{8 x-24 y}{4} $$

If the three lengths of the sides of a triangle are known, Heron's formula can be used to find its area. If \(a, b,\) and \(c\) are the lengths of the three sides, Heron's formula for area is $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where \(s\) is half the perimeter of the triangle, or \(s=\frac{1}{2}(a+b+c)\). Use this formula to find the area of each triangle. Give an exact answer and then a two-decimal-place approximation. In your own words, explain why you think \(s\) in Heron's formula is called the semiperimeter.

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ g(7) $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[8]{x^{4} y^{4}} $$

The maximum distance \(D(h)\) in kilometers that a person can see from a height h kilometers above the ground is given by the func\(\operatorname{tion} D(h)=111.7 \sqrt{h} .\) Find the height that would allow a person to see 80 kilometers.

Perform each indicated operation. Write the result in the form \(a+b i\). $$ 4(2-i)^{2} $$