Chapter 7

A spotlight is mounted on the eaves of a house 12 feet above the ground. A flower bed runs between the house and the sidewalk, so the closest a ladder can be placed to the house is 5 feet. How long of a ladder is needed so that an electrician can reach the place where the light is mounted?

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (8-3 i)+(2+3 i) $$

Multiply and then simplify if possible. $$ (2 \sqrt{7}+3 \sqrt{5})(\sqrt{7}-2 \sqrt{5}) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{9 x^{4} y^{6}} $$

Multiply. $$ \frac{\left(2 x^{1 / 5}\right)^{4}}{x^{3 / 10}} $$

A wire is to be attached to support a telephone pole. Because of surrounding buildings, sidewalks, and roadways, the wire must be anchored exactly 15 feet from the base of the pole. Telephone company workers have only 30 feet of cable, and 2 feet of that must be used to attach the cable to the pole and to the stake on the ground. How high from the base of the pole can the wire be attached?

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (7+4 i)+(4-4 i) $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt{45}}{\sqrt{9}} $$

Multiply and then simplify if possible. $$ (\sqrt{6}-4 \sqrt{2})(3 \sqrt{6}+\sqrt{2}) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[3]{-27 x^{12} y^{9}} $$

Multiply. $$ x^{2 / 3}(x-2) $$

The radius of the moon is 1080 miles. Use the formula for the radius \(r\) of a sphere given its surface area \(A\). $$r=\sqrt{\frac{A}{4 \pi}}$$ to find the surface area of the moon. Round to the nearest square mile. (Source: National Space Science Data Center)

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (1-i)(1+i) $$

Multiply and then simplify if possible. $$ (\sqrt{x}-y)(\sqrt{x}+y) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[3]{-8 a^{21} b^{6}} $$

Multiply. $$ 3 x^{1 / 2}(x+y) $$

Police departments find it very useful to be able to approximate the speed of a car when they are given the distance that the car skidded before it came to a stop. If the road surface is wet concrete, the function \(S(x)=\sqrt{10.5 x}\) is used, where \(S(x)\) is the speed of the car in miles per hour and \(x\) is the distance skidded in feet. Find how fast a car was moving if it skidded 280 feet on wet concrete.

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

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

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

Multiply. $$ \left(2 x^{1 / 3}+3\right)\left(2 x^{1 / 3}-3\right) $$

The formula \(v=\sqrt{2 g h}\) gives the velocity \(v,\) in feet per second, of an object when it falls \(h\) feet accelerated by gravity \(g,\) in feet per second squared. If \(g\) is approximately 32 feet per second squared, find how far an object has fallen if its velocity is 80 feet per second.

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{16+15 i}{-3 i} $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{5 \sqrt[4]{48}}{\sqrt[4]{3}} $$

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

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{x^{8} y^{12}} $$

Multiply. $$ \left(y^{1 / 2}+5\right)\left(y^{1 / 2}+5\right) $$

Two tractors are pulling a tree stump from a field. If two forces \(A\) and \(B\) pull at right angles \(\left(90^{\circ}\right)\) to each other, the size of the resulting force \(R\) is given by the formula \(R=\sqrt{A^{2}+B^{2}} .\) If tractor \(A\) is exerting 600 pounds of force and the resulting force is 850 pounds, find how much force tractor \(B\) is exerting.

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{2-3 i}{-7 i} $$

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

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[5]{-32 x^{10} y^{5}} $$

Factor the given factor from the expression. $$ x^{8 / 3} ; x^{8 / 3}+x^{10 / 3} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (9+8 i)^{2} $$

Multiply and then simplify if possible. $$ (\sqrt{5 x}-2 \sqrt{3 x})(\sqrt{5 x}-3 \sqrt{3 x}) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[5]{-243 x^{5} z^{15}} $$

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

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (4-7 i)^{2} $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt{a^{7} b^{6}}}{\sqrt{a^{3} b^{2}}} $$

Multiply and then simplify if possible. $$ (5 \sqrt{7 x}-\sqrt{2 x})(4 \sqrt{7 x}+6 \sqrt{2 x}) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{25}{49}} $$

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

Find the period of a pendulum whose length is 2 feet. Give an exact answer and a two-decimal-place approximation.

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

Multiply and then simplify if possible. $$ (\sqrt[3]{4}+2)(\sqrt[3]{2}-1) $$

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{8 \sqrt[3]{54 m^{7}}}{\sqrt[3]{2 m}} $$

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

Factor the given factor from the expression. $$ x^{27} ; x^{3 / 7}-2 x^{27} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{5}{3-2 i} $$

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

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{x^{20}}{4 y^{2}}} $$