91Ó°ÊÓ

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ (-8)^{7 / 3} $$

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ (-27)^{4 / 3} $$

Suppose \(a \neq 0\) and \(b^{2} \geq 4 a c\). Verify by direct substitution that if $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} $$ then \(a x^{2}+b x+c=0\).

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{9} $$

Suppose \(a \neq 0\) and \(b^{2} \geq 4 a c .\) Verify by direct calculation that $$ \begin{array}{l} a x^{2}+b x+c= \\ a\left(x-\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}\right)\left(x-\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}\right) \end{array} $$.

Suppose \(f(x)=a x^{2}+b x+c,\) where \(a \neq 0 .\) Show that the vertex of the graph of \(f\) is the point \(\left(-\frac{b}{2 a}, \frac{4 a c-b^{2}}{4 a}\right)\).

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{12} $$

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{1 / 7} $$

Suppose \(b\) and \(c\) are numbers such that the equation $$ x^{2}+b x+c=0 $$ has no real solutions. Explain why the equation $$ x^{2}+b x-c=0 $$ has two real solutions.

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{1 / 11} $$