Exam 7: Radical Expressions and Equations

An oil platform is 7 miles off the coast, and a pipeline needs to run to a refinery that is 15 miles north of the platform along the coast. This requires that some of the pipeline will be offshore and some will run on land along the coast. If the pipeline comes on shore $x$ miles north of the platform, the underwater distance is given by $f(x)=\sqrt{49+x^{2}}$ , and the distance on land is given by $g(x)=15$ - $\mathrm{x}$ . The cost of the underwater pipeline is $\$ 500,000$ per mile, and the cost of the pipeline on land is $\$ 350,000$ per mile. If the pipeline comes on shore 6 miles north of the platform, what is the total cost of the pipeline? Do not round until the final step and then round to the nearest cent.

Simplify by rationalizing the denominator. Write your answer in a + bi form. - $\frac{6+5 i}{8+9 i}$

The velocity $\mathrm{v}$ in feet per second of a free- falling object can be found using the formula $\mathrm{v}=\sqrt{64 \mathrm{~d}}$ , where $\mathrm{d}$ is the distance (in feet) that the object has already fallen. Find the velocity of an object that has already fallen 210 feet. Round to the nearest foot per second.

Simplify the radical expression. Assume all variables represent nonnegative real numbers. - $\sqrt{112 x^{2} y}$

Multiply. Write your answer in a +bi form. - $(4+2 \mathrm{i})(5-7 \mathrm{i})$

The radius, $r$ , of a right circular cone can be found using the formula $r=\sqrt{\frac{3 V}{\pi h}}$ , where $V$ is the volume and $\mathrm{h}$ is the height. If the volume is 680 cubic inches and the height is 6 inches, what is the radius? (Use 3.14 for $\pi$ , and round your answer to the nearest tenth of an inch.)

Solve. Check for extraneous solutions. - $\sqrt[3]{2 \mathrm{x}-1}=\sqrt[3]{3 \mathrm{x}+2}$

Multiply - $(\sqrt{13}+\sqrt{3})(\sqrt{3}+\sqrt{2})$

Expressions of the form $a^{m / n}$ can be rewritten as $(\sqrt[n]{a})^{m}$ or as $\sqrt[n]{a^{m}}$ . Which radical expression would you use when simplifying $36^{3 / 2}$ ? Why?

A pendulum has a length of 2.6 feet. Find its period. Use the formula $T=2 \pi \sqrt{\frac{L}{32}}$ .

If $n$ is odd, under what condition is $\sqrt[n]{a}$ negative?

Add or subtract. Assume all variables represent nonnegative real numbers. - $6 \sqrt[11]{\mathrm{x}^{12} \mathrm{y}}-5 \sqrt[11]{\mathrm{xy}}$

Rewrite as a radical expression and simplify if possible. Assume all variables represent nonnegative real numbers. - $\left(y^{20}\right)^{1 / 4}$

A ssume all variables represent nonnegative real numbers. - $\sqrt{14 m^{5}} \sqrt{7 m^{5}}$

Tell whether the statement is true or false. If it is false, give the orrect answer. Explain your answer. The easiest way to simplify the expression $\frac{2}{2+\sqrt{2}}$ is to multiply both the numerator and denominator by $2-\sqrt{2}$ .

Find the domain of the function. Express the answer in interval notation. - $f(x)=3-5 \sqrt[4]{3 t+30}$

Rationalize the denominator and simplify. A ssume all variables represent nonnegative real numbers. - $\sqrt[3]{\frac{7}{9 x^{2}}}$

Approximate to the nearest thousandth, using a calculator - $\sqrt{417}$