Chapter 10

A wire is to be attached to support a telephone pole. Because of surrounding buildings, sidewalks, and roadway, 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?

Assume that all variables represent positive real numbers. $$ -\sqrt[3]{125} $$

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

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{45}}{\sqrt{9}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{x}-y)(\sqrt{x}+y) $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{6+2 i}{4-3 i} $$

When rationalizing the denominator of \(\frac{\sqrt{5}}{\sqrt{7}}\), explain why both the numerator and the denominator must be multiplied by \(\sqrt{7}\).

Assume that all variables represent positive real numbers. $$ \sqrt[3]{8 x^{3}} $$

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

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{24}}{\sqrt[3]{3}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{6}-4 \sqrt{2})(3 \sqrt{6}+1) $$

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)

Divide. Write your answers in the form \(a+b i\) $$ \frac{4-5 i}{2 i} $$

When rationalizing the numerator of \(\frac{\sqrt{5}}{\sqrt{7}},\) explain why both the numerator and the denominator must be multiplied by \(\sqrt{5}\).

Police departments find it very useful to be able to approximate driving speeds in skidding accidents. 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.

Assume that all variables represent positive real numbers. $$ \sqrt{16 x^{8}} $$

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

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{10}}{\sqrt[3]{2}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (3 \sqrt{x}+2)(\sqrt{3 x}-2) $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{6+8 i}{3 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{2-\sqrt{11}}{6}\)

The formula \(v=\sqrt{2 g h}\) relates the velocity \(v\), in feet per second, of an object after 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.

Assume that all variables represent positive real numbers. $$ \sqrt{y^{12}} $$

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

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{5 \sqrt[4]{48}}{\sqrt[4]{3}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{3}+x)^{2} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{16+15 i}{-3 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{15}+1}{2}\)

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 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.

Assume that all variables represent positive real numbers. $$ \sqrt[3]{y^{12}} $$

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

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{7 \sqrt[4]{162}}{\sqrt[4]{2}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{y}-3 x)^{2} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{2-3 i}{-7 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{2-\sqrt{7}}{-5}\)

In psychology, it has been suggested that the number S of nonsense syllables that a person can repeat consecutively depends on his or her IQ score I according to the equation \(S=2 \sqrt{I}-9\). Use this relationship to estimate the IQ of a person who can repeat 11 nonsense syllables consecutively.

Assume that all variables represent positive real numbers. $$ \sqrt{25 a^{2} b^{20}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{(x+3)^{2}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{x^{5} y^{3}}}{\sqrt{x y}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{5 x}-3 \sqrt{2})(\sqrt{5} x-3 \sqrt{3}) $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{2}{3+i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{5}+2}{\sqrt{2}}\)

In psychology, it has been suggested that the number S of nonsense syllables that a person can repeat consecutively depends on his or her IQ score I according to the equation \(S=2 \sqrt{I}-9\). Use this relationship to estimate the IQ of a person who can repeat 15 nonsense syllables consecutively.

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

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[8]{(y+1)^{4}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{a^{7} b^{6}}}{\sqrt{a^{3} b^{2}}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (5 \sqrt{3 x}-\sqrt{y})(4 \sqrt{x}+1) $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{5}{3-2 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{x}+3}{\sqrt{x}}\)

The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period \(P\), in seconds, is \(P=2 \pi \sqrt{\frac{l}{32}},\) where l is the length of the pendulum in feet. Use this formula for Exercises 69 through \(74 .\) Find the period of a pendulum whose length is 2 feet. Give an exact answer and a two-decimal-place approximation.