Solve 0.190-0.268/sqrt{(frac{frac{65{325}*260/325}{284})+(65/325*frac{26/325}{41})}{(frac{35}{325})}}

$$\frac { 0.190 - 0.268 } { \sqrt { ( \frac { \frac { 65 } { 325 } \times \frac { 260 } { 325 } } { 284 } ) + ( \frac { \frac { 65 } { 325 } \times \frac { 26 } { 325 } } { 41 } ) } { ( \frac { 35 } { 325 } ) } }$$

Evaluate

$$-\frac{507\sqrt{5050585}}{48580}\approx -23.45423602$$

Solution Steps

Subtract 0.268 from 0.19 to get -0.078.

$$\frac{-0.078}{\sqrt{\frac{\frac{65}{325}\times \frac{260}{325}}{284}+\frac{\frac{65}{325}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Reduce the fraction \frac{65}{325} to lowest terms by extracting and canceling out 65.

$$\frac{-0.078}{\sqrt{\frac{\frac{1}{5}\times \frac{260}{325}}{284}+\frac{\frac{65}{325}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Reduce the fraction \frac{260}{325} to lowest terms by extracting and canceling out 65.

$$\frac{-0.078}{\sqrt{\frac{\frac{1}{5}\times \frac{4}{5}}{284}+\frac{\frac{65}{325}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Multiply \frac{1}{5} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-0.078}{\sqrt{\frac{\frac{1\times 4}{5\times 5}}{284}+\frac{\frac{65}{325}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Do the multiplications in the fraction \frac{1\times 4}{5\times 5}.

$$\frac{-0.078}{\sqrt{\frac{\frac{4}{25}}{284}+\frac{\frac{65}{325}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Express \frac{\frac{4}{25}}{284} as a single fraction.

$$\frac{-0.078}{\sqrt{\frac{4}{25\times 284}+\frac{\frac{65}{325}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Multiply 25 and 284 to get 7100.

$$\frac{-0.078}{\sqrt{\frac{4}{7100}+\frac{\frac{65}{325}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Reduce the fraction \frac{4}{7100} to lowest terms by extracting and canceling out 4.

$$\frac{-0.078}{\sqrt{\frac{1}{1775}+\frac{\frac{65}{325}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Reduce the fraction \frac{65}{325} to lowest terms by extracting and canceling out 65.

$$\frac{-0.078}{\sqrt{\frac{1}{1775}+\frac{\frac{1}{5}\times \frac{26}{325}}{41}}\times \frac{35}{325}}$$

Reduce the fraction \frac{26}{325} to lowest terms by extracting and canceling out 13.

$$\frac{-0.078}{\sqrt{\frac{1}{1775}+\frac{\frac{1}{5}\times \frac{2}{25}}{41}}\times \frac{35}{325}}$$

Multiply \frac{1}{5} times \frac{2}{25} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-0.078}{\sqrt{\frac{1}{1775}+\frac{\frac{1\times 2}{5\times 25}}{41}}\times \frac{35}{325}}$$

Do the multiplications in the fraction \frac{1\times 2}{5\times 25}.

$$\frac{-0.078}{\sqrt{\frac{1}{1775}+\frac{\frac{2}{125}}{41}}\times \frac{35}{325}}$$

Express \frac{\frac{2}{125}}{41} as a single fraction.

$$\frac{-0.078}{\sqrt{\frac{1}{1775}+\frac{2}{125\times 41}}\times \frac{35}{325}}$$

Multiply 125 and 41 to get 5125.

$$\frac{-0.078}{\sqrt{\frac{1}{1775}+\frac{2}{5125}}\times \frac{35}{325}}$$

Least common multiple of 1775 and 5125 is 363875. Convert \frac{1}{1775} and \frac{2}{5125} to fractions with denominator 363875.

$$\frac{-0.078}{\sqrt{\frac{205}{363875}+\frac{142}{363875}}\times \frac{35}{325}}$$

Since \frac{205}{363875} and \frac{142}{363875} have the same denominator, add them by adding their numerators.

$$\frac{-0.078}{\sqrt{\frac{205+142}{363875}}\times \frac{35}{325}}$$

Add 205 and 142 to get 347.

$$\frac{-0.078}{\sqrt{\frac{347}{363875}}\times \frac{35}{325}}$$

Rewrite the square root of the division \sqrt{\frac{347}{363875}} as the division of square roots \frac{\sqrt{347}}{\sqrt{363875}}.

$$\frac{-0.078}{\frac{\sqrt{347}}{\sqrt{363875}}\times \frac{35}{325}}$$

Factor 363875=5^{2}\times 14555. Rewrite the square root of the product \sqrt{5^{2}\times 14555} as the product of square roots \sqrt{5^{2}}\sqrt{14555}. Take the square root of 5^{2}.

$$\frac{-0.078}{\frac{\sqrt{347}}{5\sqrt{14555}}\times \frac{35}{325}}$$

Rationalize the denominator of \frac{\sqrt{347}}{5\sqrt{14555}} by multiplying numerator and denominator by \sqrt{14555}.

$$\frac{-0.078}{\frac{\sqrt{347}\sqrt{14555}}{5\left(\sqrt{14555}\right)^{2}}\times \frac{35}{325}}$$

The square of \sqrt{14555} is 14555.

$$\frac{-0.078}{\frac{\sqrt{347}\sqrt{14555}}{5\times 14555}\times \frac{35}{325}}$$

To multiply \sqrt{347} and \sqrt{14555}, multiply the numbers under the square root.

$$\frac{-0.078}{\frac{\sqrt{5050585}}{5\times 14555}\times \frac{35}{325}}$$

Multiply 5 and 14555 to get 72775.

$$\frac{-0.078}{\frac{\sqrt{5050585}}{72775}\times \frac{35}{325}}$$

Reduce the fraction \frac{35}{325} to lowest terms by extracting and canceling out 5.

$$\frac{-0.078}{\frac{\sqrt{5050585}}{72775}\times \frac{7}{65}}$$

Multiply \frac{\sqrt{5050585}}{72775} times \frac{7}{65} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-0.078}{\frac{\sqrt{5050585}\times 7}{72775\times 65}}$$

Divide -0.078 by \frac{\sqrt{5050585}\times 7}{72775\times 65} by multiplying -0.078 by the reciprocal of \frac{\sqrt{5050585}\times 7}{72775\times 65}.

$$\frac{-0.078\times 72775\times 65}{\sqrt{5050585}\times 7}$$

Rationalize the denominator of \frac{-0.078\times 72775\times 65}{\sqrt{5050585}\times 7} by multiplying numerator and denominator by \sqrt{5050585}.

$$\frac{-0.078\times 72775\times 65\sqrt{5050585}}{\left(\sqrt{5050585}\right)^{2}\times 7}$$

The square of \sqrt{5050585} is 5050585.

$$\frac{-0.078\times 72775\times 65\sqrt{5050585}}{5050585\times 7}$$

Multiply -0.078 and 72775 to get -5676.45.

$$\frac{-5676.45\times 65\sqrt{5050585}}{5050585\times 7}$$

Multiply -5676.45 and 65 to get -368969.25.

$$\frac{-368969.25\sqrt{5050585}}{5050585\times 7}$$

Multiply 5050585 and 7 to get 35354095.

$$\frac{-368969.25\sqrt{5050585}}{35354095}$$

Divide -368969.25\sqrt{5050585} by 35354095 to get -\frac{507}{48580}\sqrt{5050585}.

$$-\frac{507}{48580}\sqrt{5050585}$$