Chapter 7

Write in the form \(a+b i\). $$ \frac{4-\sqrt{-8}}{2} $$

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \sqrt[3]{7} \cdot \sqrt[3]{11}=\sqrt[3]{18} $$

Explain why \(\sqrt{-64}\) is not a real number.

Write in the form \(a+b i\). $$ \frac{5-\sqrt{-75}}{10} $$

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \sqrt[3]{7} \cdot \sqrt{11}=\sqrt{77} $$

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ -8^{1 / 3} $$

Explain why \(\sqrt[3]{-64}\) is a real number.

Write in the form \(a+b i\). $$ \frac{7+\sqrt{-98}}{14} $$

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \sqrt{x^{7} y^{8}}=\sqrt{x^{7}} \cdot \sqrt{y^{8}} $$

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ (-8)^{1 / 3} $$

The Mosteller formula for calculating adult body surface area is \(B=\sqrt{\frac{h w}{3131}},\) where \(B\) is an individual's body surface area in square meters, \(h\) is the individual's height in inches, and \(w\) is the individual's weight in pounds. Use this information to answer Exercises 113 and 114 . Round answers to 2 decimal places. Find the body surface area of an individual who is 66 inches tall and who weighs 135 pounds.

Basal metabolic rate \((B M R)\) is the number of calories per day a person needs to maintain life. A person's basal metabolic rate \(B(w)\) in calories per day can be estimated with the function \(B(w)=70 w^{3 / 4},\) where \(w\) is the person's weight in kilograms. Use this information to answer Estimate the BMR for a person who weighs 60 kilograms. Round to the nearest calorie. (Note: 60 kilograms is approximately 132 pounds.)

Write in the form \(a+b i\). Describe how to find the conjugate of a complex number.

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} $$

The Mosteller formula for calculating adult body surface area is \(B=\sqrt{\frac{h w}{3131}},\) where \(B\) is an individual's body surface area in square meters, \(h\) is the individual's height in inches, and \(w\) is the individual's weight in pounds. Use this information to answer Exercises 113 and 114 . Round answers to 2 decimal places. Find the body surface area of an individual who is 74 inches tall and who weighs 225 pounds.

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \frac{\sqrt[3]{12}}{\sqrt[3]{4}}=\sqrt[3]{8} $$

Write in the form \(a+b i\). Explain why the product of a complex number and its comSlex conjugate is a real number.

Escape velocity is the minimum speed that an object must reach to escape the pull of a planet's gravity. Escape velocity \(v\) is given by the equation \(v=\sqrt{\frac{2 G m}{r}},\) where \(m\) is the mass of the planet, \(r\) is its radius, and \(G\) is the universal gravitational constant, which has a value of \(G=6.67 \times 10^{-11} \mathrm{m}^{3} / \mathrm{kg} \cdot \mathrm{s}^{2} .\) The mass of Earth is \(5.97 \times 10^{24} \mathrm{kg},\) and its radius is \(6.37 \times 10^{6} \mathrm{m} .\) Use this information to find the escape velocity for Earth in meters per second. Round to the nearest whole number. (Source: National Space Science Data Center)

The number of cell telephone subscribers in the United States from \(1995-2010\) can be modeled by \(f(x)=25 x^{23 / 25},\) where \(f(x)\) is the number of cellular telephone subscriptions in millions, \(x\) years after \(1995 . \text { (Source: CTIA-Wireless Association, } 1995-2010)\) Use this information to answer. Use this model to estimate the number of cellular subscriptions in \(2010 .\) Round to the nearest tenth of a million.

Find and correct the error. See the Concept Check in this section. $$ \frac{\sqrt[3]{64}}{\sqrt{64}}=\sqrt[3]{\frac{64}{64}}=\sqrt[3]{1}=1 $$

The number of cell telephone subscribers in the United States from \(1995-2010\) can be modeled by \(f(x)=25 x^{23 / 25},\) where \(f(x)\) is the number of cellular telephone subscriptions in millions, \(x\) years after \(1995 . \text { (Source: CTIA-Wireless Association, } 1995-2010)\) Use this information to answer. Predict the number of cellular telephone subscriptions in \(2015 .\) Round to the nearest tenth of a million.

Simplify. $$ (8-\sqrt{-4})-(2+\sqrt{-16}) $$

Find and correct the error. See the Concept Check in this section. $$ \frac{\sqrt[4]{16}}{\sqrt{4}}=\sqrt[4]{\frac{16}{4}}=\sqrt[4]{4} $$

Suppose a classmate tells you that \(\sqrt{13} \approx 5.7 .\) Without a calculator, how can you convince your classmate that he or she must have made an error?

The number of cell telephone subscribers in the United States from \(1995-2010\) can be modeled by \(f(x)=25 x^{23 / 25},\) where \(f(x)\) is the number of cellular telephone subscriptions in millions, \(x\) years after \(1995 . \text { (Source: CTIA-Wireless Association, } 1995-2010)\) Use this information to answer. Explain how writing \(x^{-7}\) with positive exponents is similar to writing \(x^{-1 / 4}\) with positive exponents.

Simplify. Determine whether \(2 i\) is a solution of \(x^{2}+4=0\)

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[5]{x^{35}} $$

Suppose a classmate tells you that \(\sqrt[3]{10} \approx 3.2 .\) Without a calculator, how can you convince your friend that he or she must have made an error?

The number of cell telephone subscribers in the United States from \(1995-2010\) can be modeled by \(f(x)=25 x^{23 / 25},\) where \(f(x)\) is the number of cellular telephone subscriptions in millions, \(x\) years after \(1995 . \text { (Source: CTIA-Wireless Association, } 1995-2010)\) Use this information to answer. Explain how writing \(2 x^{-5}\) with positive exponents is similar to writing \(2 x^{-3 / 4}\) with positive exponents.

Simplify. Determine whether \(-1+i\) is a solution of \(x^{2}+2 x=-2\)

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[6]{y^{48}} $$

Fill in each box with the correct expression. $$ \square \cdot a^{2 / 3}=a^{3 / 3}, \text { or } a $$

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[4]{a^{12} b^{4} c^{20}} $$

Fill in each box with the correct expression. $$ \square \cdot x^{1 / 8}=x^{4 / 8}, \text { or } x^{1 / 2} $$

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[3]{a^{9} b^{21} c^{3}} $$

Fill in each box with the correct expression. $$ \frac{\square}{x^{-2 / 5}}=x^{3 / 5} $$

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[3]{z^{32}} $$

Fill in each box with the correct expression. $$ \frac{\square}{y^{-3 / 4}}=y^{4 / 4}, \text { or } y $$

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[5]{x^{49}} $$

Use a calculator to write a four-decimal-place approximation of each number. $$ 8^{1 / 4} $$

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[7]{q^{17} r^{40} s^{7}} $$

Use a calculator to write a four-decimal-place approximation of each number. $$ 20^{1 / 5} $$

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[4]{p^{11} q^{4} r^{45}} $$

Use a calculator to write a four-decimal-place approximation of each number. $$ 18^{3 / 5} $$

The formula for the radius \(r\) of a sphere with surface area \(A\) is given by \(r=\sqrt{\frac{A}{4 \pi}}\). Calculate the radius of a standard zorb whose outside surface area is 32.17 sq \(\mathrm{m}\). Round to the nearest tenth. (A zorb is a large inflated ball within a ball in which a person, strapped inside, may choose to roll down a hill. Source: Zorb, Ltd.) (IMAGE CANNOT COPY)

Use a calculator to write a four-decimal-place approximation of each number. $$ 76^{5 \sqrt{7}} $$

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. The owner of Knightime Classic Movie Rentals has determined that the demand equation for renting older released DVDs is \(F(x)=0.6 \sqrt{49-x^{2}},\) where \(x\) is the price in dollars per two-day rental and \(F(x)\) is the number of times the \(\mathrm{DVD}\) is demanded per week. A. Approximate to one decimal place the demand per week of an older released DVD if the rental price is S3 per two-day rental. B. Approximate to one decimal place the demand per week of an older released DVD if the rental price is S5 per two-day rental. C. Explain how the owner of the store can use this equation to predict the number of copies of each DVD that should be in stock.

In physics, the speed of a wave traveling over a stretched string with tension \(t\) and density \(u\) is given by the expression \(\frac{\sqrt{t}}{\sqrt{u}} .\) Write this expression with rational exponents.

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. The formula for the lateral surface area \(A\) of a cone with height \(h\) and radius \(r\) is given by $$ A=\pi r \sqrt{r^{2}+h^{2}} $$ a. Find the lateral surface area of a cone whose height is 3 centimeters and whose radius is 4 centimeters. b. Approximate to two decimal places the lateral surface area of a cone whose height is 7.2 feet and whose radius is 6.8 feet.

In electronics, the angular frequency of oscillations in a certain type of circuit is given by the expression \((L C)^{-1 / 2} .\) Use radical notation to write this expression.