Solve (-1/2+5/3)^-1:6/1-sqrt{1/1-frac{7}{8}}+frac{2}{1}*(-frac{7}{1})^-1=

$$( - \frac { 1 } { 2 } + \frac { 5 } { 3 } ) ^ { - 1 } : \frac { 6 } { 1 } - \sqrt { \frac { 1 } { 1 } - \frac { 7 } { 8 } } + \frac { 2 } { 1 } \cdot ( - \frac { 7 } { 1 } ) ^ { - 1 } =$$

Evaluate

$$-\frac{\sqrt{2}}{4}-\frac{1}{7}\approx -0.496410533$$

Short Solution Steps

Divide 1 by 1 to get 1.

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

Add -\frac{1}{2} and \frac{5}{3} to get \frac{7}{6}.

$$\frac{\left(\frac{7}{6}\right)^{-1}}{\frac{6}{1}}-\sqrt{1-\frac{7}{8}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Calculate \frac{7}{6} to the power of -1 and get \frac{6}{7}.

$$\frac{\frac{6}{7}}{\frac{6}{1}}-\sqrt{1-\frac{7}{8}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Anything divided by one gives itself.

$$\frac{\frac{6}{7}}{6}-\sqrt{1-\frac{7}{8}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Express \frac{\frac{6}{7}}{6} as a single fraction.

$$\frac{6}{7\times 6}-\sqrt{1-\frac{7}{8}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Cancel out 6 in both numerator and denominator.

$$\frac{1}{7}-\sqrt{1-\frac{7}{8}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Subtract \frac{7}{8} from 1 to get \frac{1}{8}.

$$\frac{1}{7}-\sqrt{\frac{1}{8}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$\frac{1}{7}-\frac{\sqrt{1}}{\sqrt{8}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{7}-\frac{1}{\sqrt{8}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{1}{7}-\frac{1}{2\sqrt{2}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Rationalize the denominator of \frac{1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{1}{7}-\frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

The square of \sqrt{2} is 2.

$$\frac{1}{7}-\frac{\sqrt{2}}{2\times 2}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Multiply 2 and 2 to get 4.

$$\frac{1}{7}-\frac{\sqrt{2}}{4}+\frac{2}{1}\left(-\frac{7}{1}\right)^{-1}$$

Anything divided by one gives itself.

$$\frac{1}{7}-\frac{\sqrt{2}}{4}+2\left(-\frac{7}{1}\right)^{-1}$$

Anything divided by one gives itself.

$$\frac{1}{7}-\frac{\sqrt{2}}{4}+2\left(-7\right)^{-1}$$

Calculate -7 to the power of -1 and get -\frac{1}{7}.

$$\frac{1}{7}-\frac{\sqrt{2}}{4}+2\left(-\frac{1}{7}\right)$$

Multiply 2 and -\frac{1}{7} to get -\frac{2}{7}.

$$\frac{1}{7}-\frac{\sqrt{2}}{4}-\frac{2}{7}$$

Subtract \frac{2}{7} from \frac{1}{7} to get -\frac{1}{7}.

$$-\frac{1}{7}-\frac{\sqrt{2}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 4 is 28. Multiply -\frac{1}{7} times \frac{4}{4}. Multiply \frac{\sqrt{2}}{4} times \frac{7}{7}.

$$-\frac{4}{28}-\frac{7\sqrt{2}}{28}$$

Since -\frac{4}{28} and \frac{7\sqrt{2}}{28} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4-7\sqrt{2}}{28}$$

Factor

$$\frac{-7 \sqrt{2} - 4}{28} = -0.49641053345041664$$