Solve 85.3-78.2/sqrt{frac{6.5{8}+6.3/8}}

$$\frac{ 85.3-78.2 }{ \sqrt{ \frac{ 6.5 }{ 8 } + \frac{ 6.3 }{ 8 } } }$$

Evaluate

$$\frac{71\sqrt{10}}{40}\approx 5.613042847$$

Solution Steps

Subtract 78.2 from 85.3 to get 7.1.

$$\frac{7.1}{\sqrt{\frac{6.5}{8}+\frac{6.3}{8}}}$$

Expand \frac{6.5}{8} by multiplying both numerator and the denominator by 10.

$$\frac{7.1}{\sqrt{\frac{65}{80}+\frac{6.3}{8}}}$$

Reduce the fraction \frac{65}{80} to lowest terms by extracting and canceling out 5.

$$\frac{7.1}{\sqrt{\frac{13}{16}+\frac{6.3}{8}}}$$

Expand \frac{6.3}{8} by multiplying both numerator and the denominator by 10.

$$\frac{7.1}{\sqrt{\frac{13}{16}+\frac{63}{80}}}$$

Least common multiple of 16 and 80 is 80. Convert \frac{13}{16} and \frac{63}{80} to fractions with denominator 80.

$$\frac{7.1}{\sqrt{\frac{65}{80}+\frac{63}{80}}}$$

Since \frac{65}{80} and \frac{63}{80} have the same denominator, add them by adding their numerators.

$$\frac{7.1}{\sqrt{\frac{65+63}{80}}}$$

Add 65 and 63 to get 128.

$$\frac{7.1}{\sqrt{\frac{128}{80}}}$$

Reduce the fraction \frac{128}{80} to lowest terms by extracting and canceling out 16.

$$\frac{7.1}{\sqrt{\frac{8}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{8}{5}} as the division of square roots \frac{\sqrt{8}}{\sqrt{5}}.

$$\frac{7.1}{\frac{\sqrt{8}}{\sqrt{5}}}$$

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}.

$$\frac{7.1}{\frac{2\sqrt{2}}{\sqrt{5}}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{7.1}{\frac{2\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{7.1}{\frac{2\sqrt{2}\sqrt{5}}{5}}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{7.1}{\frac{2\sqrt{10}}{5}}$$

Divide 7.1 by \frac{2\sqrt{10}}{5} by multiplying 7.1 by the reciprocal of \frac{2\sqrt{10}}{5}.

$$\frac{7.1\times 5}{2\sqrt{10}}$$

Rationalize the denominator of \frac{7.1\times 5}{2\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{7.1\times 5\sqrt{10}}{2\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{7.1\times 5\sqrt{10}}{2\times 10}$$

Multiply 7.1 and 5 to get 35.5.

$$\frac{35.5\sqrt{10}}{2\times 10}$$

Multiply 2 and 10 to get 20.

$$\frac{35.5\sqrt{10}}{20}$$

Divide 35.5\sqrt{10} by 20 to get 1.775\sqrt{10}.

$$1.775\sqrt{10}$$