91Ó°ÊÓ

Use the special products of this section to determine the products. Each comes from the technical area indicated. \(\frac{1}{2} \pi(R+r)(R-r) \quad\) (architecture)

In Exercises \(65-68,\) after finding the simplest form of each fraction, explain why it cannot be simplified more. (a) \(\frac{x^{2}-x-2}{x^{2}-x}\) (b) \(\frac{x^{2}-x-2}{x^{2}+x}\)

Perform the indicated operations. Each expression occurs in the indicated area of application. $$\frac{1}{R^{2}}+\left(\omega C-\frac{1}{\omega L}\right)^{2} \quad \text { (electricity) }$$

Factor the given expressions completely. Each is from the technical area indicated. $$k^{2} A^{2}+2 k \lambda A+\lambda^{2}-\alpha^{2} \quad \text { (robotics) }$$

After finding the simplest form of each fraction, -explain why it cannot be simplified more. $$\text { (a) } \frac{x^{3}-x}{1-x} \quad \text { (b) } \frac{2 x^{2}+4 x}{2 x^{2}+4}$$

Perform the indicated operations. Each expression occurs in the indicated area of application. $$\frac{\frac{x}{h_{1}}+\frac{x-L}{h_{2}}}{1+\frac{x(L-x)}{h_{1} h_{2}}} \text { (optics) }$$

In Exercises \(65-68,\) after finding the simplest form of each fraction, explain why it cannot be simplified more. (a) \(\frac{x^{3}-x}{1-x}\) (b) \(\frac{2 x^{2}+4 x}{2 x^{2}+4}\)

Factor the expressions completely. It is necessary to set up the proper expression. Each expression comes from the technical area indicated. \(p_{1} R^{2}-p_{1} r^{2}-p_{2} R^{2}+p_{2} r^{2} \quad\) (fluid flow)

Give the required explanations.Factor the expressions completely. \(p_{1} R^{2}-p_{1} r^{2}-p_{2} R^{2}+p_{2} r^{2} \quad\) (fluid flow)

Use the special products of this section to determine the products. Each comes from the technical area indicated. \(K\left(T-T_{0}\right)\left(T+T_{0}\right)\left(T^{2}+T_{0}^{2}\right) \quad\) (radiation)