91Ó°ÊÓ

Find the midpoint of the segment with the given endpoints. $$ (\sqrt{2},-1) \text { and }(\sqrt{3}, 4) $$

Divide and simplify. Write each answer in the form \(a+b i\). $$\frac{7 i+14}{7 i}$$

For an engine with a displacement of \(2.8 \mathrm{L}\), the function given by $$ d(n)=0.75 \sqrt{2.8 n} $$ can be used to determine the diameter size of the carburetor's opening, in millimeters. Here \(n\) is the number of rpm's at which the engine achieves peak performance. If a carburetor's opening is \(81 \mathrm{mm},\) for what number of rpm's will the engine produce peak power?

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$\sqrt[5]{x^{3}(y+z)^{6}} \sqrt[5]{x^{3}(y+z)^{4}}$$

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. Thus absolute-value notation is not necessary. $$ -\sqrt{(3 t)^{2}} $$

Laws of Exponents Use the laws of exponents to simplify. Write answers using exponential notation, and do not use negative exponents in any answers. $$\left(10^{3 / 5}\right)^{2 / 5}$$

Rationalize each numerator. Assume that all variables represent positive numbers. If no calculator is available, why it is easier to approximate $$ \frac{\sqrt{2}}{2} \text { than } \frac{1}{\sqrt{2}} ? $$

Solve. $$ \left(\frac{z}{4}-5\right)^{2 / 3}=\frac{1}{25} $$

Divide and simplify. Write each answer in the form \(a+b i\). $$\frac{6 i+3}{3 i}$$

Laws of Exponents Use the laws of exponents to simplify. Write answers using exponential notation, and do not use negative exponents in any answers. $$\left(5^{5 / 4}\right)^{3 / 7}$$