Solve (frac{1+sqrt{9}}{2})^-1+(frac{1-sqrt[3]{-9+1}}{6})^2+(frac{8-sqrt[5]{32}}{3})^-3

$$( \frac { 1 + \sqrt { 9 } } { 2 } ) ^ { - 1 } + ( \frac { 1 - \sqrt[ 3 ] { - 9 + 1 } } { 6 } ) ^ { 2 } + ( \frac { 8 - \sqrt[ 5 ] { 32 } } { 3 } ) ^ { - 3 }$$

Evaluate

$$\frac{7}{8}=0.875$$

Solution Steps

Calculate the square root of 9 and get 3.

$$\left(\frac{1+3}{2}\right)^{-1}+\left(\frac{1-\sqrt[3]{-9+1}}{6}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Add 1 and 3 to get 4.

$$\left(\frac{4}{2}\right)^{-1}+\left(\frac{1-\sqrt[3]{-9+1}}{6}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Divide 4 by 2 to get 2.

$$2^{-1}+\left(\frac{1-\sqrt[3]{-9+1}}{6}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$\frac{1}{2}+\left(\frac{1-\sqrt[3]{-9+1}}{6}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Add -9 and 1 to get -8.

$$\frac{1}{2}+\left(\frac{1-\sqrt[3]{-8}}{6}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Calculate \sqrt[3]{-8} and get -2.

$$\frac{1}{2}+\left(\frac{1-\left(-2\right)}{6}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

The opposite of -2 is 2.

$$\frac{1}{2}+\left(\frac{1+2}{6}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Add 1 and 2 to get 3.

$$\frac{1}{2}+\left(\frac{3}{6}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$\frac{1}{2}+\left(\frac{1}{2}\right)^{2}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1}{2}+\frac{1}{4}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Add \frac{1}{2} and \frac{1}{4} to get \frac{3}{4}.

$$\frac{3}{4}+\left(\frac{8-\sqrt[5]{32}}{3}\right)^{-3}$$

Calculate \sqrt[5]{32} and get 2.

$$\frac{3}{4}+\left(\frac{8-2}{3}\right)^{-3}$$

Subtract 2 from 8 to get 6.

$$\frac{3}{4}+\left(\frac{6}{3}\right)^{-3}$$

Divide 6 by 3 to get 2.

$$\frac{3}{4}+2^{-3}$$

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\frac{3}{4}+\frac{1}{8}$$

Add \frac{3}{4} and \frac{1}{8} to get \frac{7}{8}.

$$\frac{7}{8}$$

Factor

$$\frac{7}{2 ^ {3}} = 0.875$$