Solve sqrt[3]{(3/5)^-1-1/6/sqrt[3]{frac{3{4}-1+1/8}*frac{10}{(frac{3}{2})^-2+frac{2}{3}}}+frac{(frac{7}{5}-frac{3}{5}-frac{1}{2})(1-frac{5}{6})^-2}{sqrt[3]{-3(-frac{1}{3})^-2}*frac{2}{3}*6}}

$$\sqrt[ 3 ] { \frac { ( \frac { 3 } { 5 } ) ^ { - 1 } - \frac { 1 } { 6 } } { \sqrt[ 3 ] { \frac { 3 } { 4 } - 1 + \frac { 1 } { 8 } } \times \frac { 10 } { ( \frac { 3 } { 2 } ) ^ { - 2 } + \frac { 2 } { 3 } } } + \frac { ( \frac { 7 } { 5 } - \frac { 3 } { 5 } - \frac { 1 } { 2 } ) ( 1 - \frac { 5 } { 6 } ) ^ { - 2 } } { \sqrt[ 3 ] { - 3 ( - \frac { 1 } { 3 } ) ^ { - 2 } } \times \frac { 2 } { 3 } \times 6 } }$$

Evaluate

$$-\frac{\sqrt[3]{37}\times 10^{\frac{2}{3}}\times 27^{\frac{8}{9}}}{270}\approx -1.072408275$$

Solution Steps

Calculate \frac{3}{5} to the power of -1 and get \frac{5}{3}.

$$\sqrt[3]{\frac{\frac{5}{3}-\frac{1}{6}}{\sqrt[3]{\frac{3}{4}-1+\frac{1}{8}}\times \frac{10}{\left(\frac{3}{2}\right)^{-2}+\frac{2}{3}}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Subtract \frac{1}{6} from \frac{5}{3} to get \frac{3}{2}.

$$\sqrt[3]{\frac{\frac{3}{2}}{\sqrt[3]{\frac{3}{4}-1+\frac{1}{8}}\times \frac{10}{\left(\frac{3}{2}\right)^{-2}+\frac{2}{3}}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Subtract 1 from \frac{3}{4} to get -\frac{1}{4}.

$$\sqrt[3]{\frac{\frac{3}{2}}{\sqrt[3]{-\frac{1}{4}+\frac{1}{8}}\times \frac{10}{\left(\frac{3}{2}\right)^{-2}+\frac{2}{3}}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

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

$$\sqrt[3]{\frac{\frac{3}{2}}{\sqrt[3]{-\frac{1}{8}}\times \frac{10}{\left(\frac{3}{2}\right)^{-2}+\frac{2}{3}}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Calculate \sqrt[3]{-\frac{1}{8}} and get -\frac{1}{2}.

$$\sqrt[3]{\frac{\frac{3}{2}}{-\frac{1}{2}\times \frac{10}{\left(\frac{3}{2}\right)^{-2}+\frac{2}{3}}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Calculate \frac{3}{2} to the power of -2 and get \frac{4}{9}.

$$\sqrt[3]{\frac{\frac{3}{2}}{-\frac{1}{2}\times \frac{10}{\frac{4}{9}+\frac{2}{3}}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Add \frac{4}{9} and \frac{2}{3} to get \frac{10}{9}.

$$\sqrt[3]{\frac{\frac{3}{2}}{-\frac{1}{2}\times \frac{10}{\frac{10}{9}}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Divide 10 by \frac{10}{9} by multiplying 10 by the reciprocal of \frac{10}{9}.

$$\sqrt[3]{\frac{\frac{3}{2}}{-\frac{1}{2}\times 10\times \frac{9}{10}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Multiply 10 and \frac{9}{10} to get 9.

$$\sqrt[3]{\frac{\frac{3}{2}}{-\frac{1}{2}\times 9}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Multiply -\frac{1}{2} and 9 to get -\frac{9}{2}.

$$\sqrt[3]{\frac{\frac{3}{2}}{-\frac{9}{2}}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Divide \frac{3}{2} by -\frac{9}{2} by multiplying \frac{3}{2} by the reciprocal of -\frac{9}{2}.

$$\sqrt[3]{\frac{3}{2}\left(-\frac{2}{9}\right)+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Multiply \frac{3}{2} and -\frac{2}{9} to get -\frac{1}{3}.

$$\sqrt[3]{-\frac{1}{3}+\frac{\left(\frac{7}{5}-\frac{3}{5}-\frac{1}{2}\right)\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Subtract \frac{3}{5} from \frac{7}{5} to get \frac{4}{5}.

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

Subtract \frac{1}{2} from \frac{4}{5} to get \frac{3}{10}.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{3}{10}\left(1-\frac{5}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Subtract \frac{5}{6} from 1 to get \frac{1}{6}.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{3}{10}\times \left(\frac{1}{6}\right)^{-2}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Calculate \frac{1}{6} to the power of -2 and get 36.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{3}{10}\times 36}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Multiply \frac{3}{10} and 36 to get \frac{54}{5}.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{54}{5}}{\sqrt[3]{-3\left(-\frac{1}{3}\right)^{-2}}\times \frac{2}{3}\times 6}}$$

Calculate -\frac{1}{3} to the power of -2 and get 9.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{54}{5}}{\sqrt[3]{-3\times 9}\times \frac{2}{3}\times 6}}$$

Multiply -3 and 9 to get -27.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{54}{5}}{\sqrt[3]{-27}\times \frac{2}{3}\times 6}}$$

Calculate \sqrt[3]{-27} and get -3.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{54}{5}}{-3\times \frac{2}{3}\times 6}}$$

Multiply -3 and \frac{2}{3} to get -2.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{54}{5}}{-2\times 6}}$$

Multiply -2 and 6 to get -12.

$$\sqrt[3]{-\frac{1}{3}+\frac{\frac{54}{5}}{-12}}$$

Express \frac{\frac{54}{5}}{-12} as a single fraction.

$$\sqrt[3]{-\frac{1}{3}+\frac{54}{5\left(-12\right)}}$$

Multiply 5 and -12 to get -60.

$$\sqrt[3]{-\frac{1}{3}+\frac{54}{-60}}$$

Reduce the fraction \frac{54}{-60} to lowest terms by extracting and canceling out 6.

$$\sqrt[3]{-\frac{1}{3}-\frac{9}{10}}$$

Subtract \frac{9}{10} from -\frac{1}{3} to get -\frac{37}{30}.

$$\sqrt[3]{-\frac{37}{30}}$$