91Ó°ÊÓ

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[3]{8 a^{6} b^{-3}}$$

The cost of installing insulation in a particular two-bedroom home is \(\$ 2400 .\) Present monthly heating costs average \(\$ 200,\) but the insulation is expected to reduce heating costs by \(10 \% .\) How many months will it take to recover the cost of the insulation?

Rationalize the numerator. $$\frac{\sqrt{a}-\sqrt{b}}{a^{2}-b^{2}}$$

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$(3+8 i)^{2}$$

Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[4]{81 r^{5} s^{8}} $$

Rationalize the numerator. $$\frac{\sqrt{b}+\sqrt{c}}{b^{2}-c^{2}}$$

A city government has approved the construction of an \(\$ 800\) million sports arena. Up to 5480 million will be raised by selling bonds that pay simple interest at a rate of \(6 \%\) annually. The remaining amount (up to \(\$ 640\) million) will be obtained by borrowing money from an insurance company at a simple interest rate of \(5 \% .\) Determine whether the arena can be financed so that the annual interest is \(\$ 42\) million.

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$\frac{1}{9-\sqrt{-4}}$$

Two children, who are 224 meters apart, start walking toward each other at the same instant at rates of \(1.5 \mathrm{m} / \mathrm{sec}\) and \(2 \mathrm{m} / \mathrm{sec},\) respectively (see the figure). (a) When will they meet? (b) How far will each have walked?

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{6-3 i}{2+7 i} $$