Chapter 9

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{8-x}-\sqrt{3 x-8}=\sqrt{x-4} $$

Finding Rates. \(\quad\) A student drove a distance of 135 miles at an average speed of 50 mph. How much faster would she have to drive on the return trip to save 30 minutes of driving time?

Simplify each expression. All variables represent positive real numbers. $$ \left(16 x^{4}\right)^{1 / 4} $$

Use a calculator to solve each problem. Round answers to the nearest tenth. Biology. Scientists will place five rats inside a clear plastic hemisphere and control the environment to study the rats' behavior. The function \(d(V)=\sqrt[3]{12\left(\frac{V}{\pi}\right)}\) gives the diameter of a hemisphere with volume \(V\). Use the function to determine the diameter of the base of the hemisphere, if each rat requires 125 cubic feet of living space.

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{2 \sqrt{x+1}}=\sqrt{16-4 x} $$

$$ \text { Simplify: }\left(i^{349}\right)^{-i^{456}} $$

Simplify each expression. All variables represent positive real numbers. $$ \left(-x^{4}\right)^{1 / 4} $$

The function \(s(g)=\sqrt[3]{\frac{g}{7.5}}\) determines how long (in feet) an edge of a cube-shaped tank must be if it is to hold \(g\) gallons of water. What dimensions should a cube-shaped aquarium have if it is to hold 1,250 gallons of water?

Analytical Geometry. The length of the perpendicular segment drawn from \((-2,2)\) to the line with equation \(2 x-4 y=4\) is given by $$ L=\frac{|2(-2)+(-4)(2)+(-4)|}{\sqrt{(2)^{2}+(-4)^{2}}} $$ Find \(L\). Express the result in simplified radical form. Then give an approximation to the nearest tenth.

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ (2 x-1)^{2 / 3}=x^{1 / 3} $$

Simplify \((2+3 i)^{-2}\) and write the result in the form \(a+b i\)

Simplify each expression. All variables represent positive real numbers. $$ -\left(8 a^{3} b^{6}\right)^{-2 / 3} $$

Use a calculator to solve each problem. Round answers to the nearest tenth. Collectibles. The effective rate of interest \(r\) earned by an investment is given by the formula \(r=\sqrt[n]{\frac{A}{P}}-1,\) where \(P\) is the initial investment that grows to value \(A\) after \(n\) years. Determine the effective rate of interest earned by a collector on a Lladró porcelain figurine purchased for \(\$ 800\) and sold for \(\$ 950\) five years later.

Simplify each expression. All variables represent positive real numbers. $$ -\left(25 s^{4} t^{6}\right)^{-3 / 2} $$

Engineering. Refer to the illustration below that shows a block connected to two walls by springs. A measure of how fast the block will oscillate when the spring system is set in motion is given by the formula \(\omega=\sqrt{\frac{k_{1}+k_{2}}{m}}\) where \(k_{1}\) and \(k_{2}\) indicate the stiffness of the springs and \(m\) is the mass of the block. Rationalize the right side and restate the formula. (IMAGE CAN'T COPY)

Simplify each expression. All variables represent positive real numbers. a. \(-125^{2 / 3}\) b. \((-125)^{2 / 3}\) c. \(-125^{-2 / 3}\) d. \(\frac{1}{(-125)^{-2 / 3}}\)

If \(x\) is any real number, that is, if \(x\) is unrestricted, then \(\sqrt{x^{2}}=x\) is not correct. Explain why.

Consider \(\frac{\sqrt{3}}{\sqrt{7}}=\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} .\) Explain why the expressions on the left side and the right side of the equation are equal.

Simplify each expression. All variables represent positive real numbers. a. \(81^{1 / 4}\) b. \(81^{-1 / 4}\) c. \(-81^{1 / 4}\) d. \(\frac{1}{81^{-1 / 4}}\)

Explain why \(\sqrt{36}\) is just \(6,\) and not also \(-6\)

To rationalize the denominator of \(\frac{\sqrt[4]{12}}{\sqrt[4]{3}},\) why wouldn't we multiply the numerator and denominator by \(\frac{\sqrt[4]{3}}{\sqrt[4]{3}} ?\)

Simplify each expression. All variables represent positive real numbers. a. \(\left(64 a^{4}\right)^{1 / 2}\) b. \(\left(64 a^{4}\right)^{-1 / 2}\) c. \(-\left(64 a^{4}\right)^{1 / 2}\) d. \(\frac{1}{\left(64 a^{4}\right)^{1 / 2}}\)

Explain why \(\frac{\sqrt[3]{12}}{\sqrt[3]{5}}\) is not in simplified form.

Simplify each expression. All variables represent positive real numbers. a. \(r^{1 / 3} \cdot r^{1 / 5}\) b. \(\left(r^{1 / 3}\right)^{1 / 5}\) c. \(r^{1 / 3} \cdot r^{-1 / 5}\) d. \(\left(r^{-1 / 3}\right)^{-1 / 5}\)

Perform the operations and simplify when possible. $$ \frac{x^{2}-3 x y-4 y^{2}}{x^{2}+c x-2 y x-2 c y} \div \frac{x^{2}-2 x y-3 y^{2}}{x^{2}+c x-4 y x-4 c y} $$

Explain why \(\sqrt{m} \cdot \sqrt{m}=m\) but \(\sqrt[3]{m} \cdot \sqrt[3]{m} \neq m .\) Assume that \(m\) represents a positive number.

Perform the operations and simplify when possible. $$ \frac{2 x+3}{3 x-1}-\frac{x-4}{2 x+1} $$

Explain why the product of \(\sqrt{m}+3\) and \(\sqrt{m}-3\) does not contain a radical.

Relativity. One concept of relativity theory is that an object moving past an observer at a speed near the speed of light appears to have a larger mass because of its motion. If the mass of the object is \(m_{0}\) when the object is at rest relative to the observer, its mass \(m\) will be given by the formula \(m=m_{0}\left(1-\frac{v^{2}}{c^{2}}\right)^{-1 / 2}\) when it is moving with speed \(v\) (in miles per second) past the observer. The variable \(c\) is the speed of light, \(186,000\) mi/sec. If a proton with a rest mass of 1 unit is accelerated by a nuclear accelerator to a speed of \(160,000 \mathrm{mi} / \mathrm{sec},\) what mass will the technicians observe it to have? Round to the nearest hundredth.

Graph \(f(x)=-\sqrt{x-2}+3\) and find the domain and range.

Solve each equation. $$ \frac{8}{b-2}+\frac{3}{2-b}=-\frac{1}{b} $$

Simplify \(\sqrt{9 a^{16}+12 a^{8} b^{25}+4 b^{50}}\) and assume that \(a>0\) and \(b>0\)

Solve each equation. $$ \frac{2}{x-2}+\frac{1}{x+1}=\frac{1}{(x+1)(x-2)} $$

Multiply: \(\quad \sqrt{2} \cdot \sqrt[3]{2} .\)

What is a rational exponent? Give some examples.

Explain how the root key \([\sqrt[x]{y}]\) on a scientific calculator can be used in combination with other keys to evaluate the expression \(16^{3 / 4}\).

Commuting Time. The time it takes a car to travel a certain distance varies inversely with its rate of speed. If a certain trip takes 3 hours at 50 miles per hour, how long will the trip take at 60 miles per hour?

Bankruptcy. After filing for bankruptcy, a company was able to pay its creditors only 15 cents on the dollar. If the company owed a lumberyard 9,712 dollars, how much could the lumberyard expect to be paid?

The fraction \(\frac{2}{4}\) is equal to \(\frac{1}{2} .\) Is \(16^{2 / 4}\) equal to \(16^{1 / 2}\) ? Explain.

Explain how would you evaluate an expression with a mixed-number exponent. For example, what is \(8^{1 \frac{1}{3}} ?\) What is \(25^{2 \frac{1}{2}} ?\)