Solve 8(frac{5/sqrt{41})-3(4/sqrt{41})}{8(5/sqrt{41})+2(4/sqrt{41})}

$$\frac{ 8( \frac{ 5 }{ \sqrt{ 41 } } )-3( \frac{ 4 }{ \sqrt{ 41 } } ) }{ 8( \frac{ 5 }{ \sqrt{ 41 } } )+2( \frac{ 4 }{ \sqrt{ 41 } } ) }$$

Evaluate

$$\frac{7}{12}\approx 0.583333333$$

Short Solution Steps

Rationalize the denominator of \frac{5}{\sqrt{41}} by multiplying numerator and denominator by \sqrt{41}.

$$\frac{8\times \frac{5\sqrt{41}}{\left(\sqrt{41}\right)^{2}}-3\times \frac{4}{\sqrt{41}}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

The square of \sqrt{41} is 41.

$$\frac{8\times \frac{5\sqrt{41}}{41}-3\times \frac{4}{\sqrt{41}}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

Express 8\times \frac{5\sqrt{41}}{41} as a single fraction.

$$\frac{\frac{8\times 5\sqrt{41}}{41}-3\times \frac{4}{\sqrt{41}}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

Rationalize the denominator of \frac{4}{\sqrt{41}} by multiplying numerator and denominator by \sqrt{41}.

$$\frac{\frac{8\times 5\sqrt{41}}{41}-3\times \frac{4\sqrt{41}}{\left(\sqrt{41}\right)^{2}}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

The square of \sqrt{41} is 41.

$$\frac{\frac{8\times 5\sqrt{41}}{41}-3\times \frac{4\sqrt{41}}{41}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

Express 3\times \frac{4\sqrt{41}}{41} as a single fraction.

$$\frac{\frac{8\times 5\sqrt{41}}{41}-\frac{3\times 4\sqrt{41}}{41}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

Multiply 3 and 4 to get 12.

$$\frac{\frac{8\times 5\sqrt{41}}{41}-\frac{12\sqrt{41}}{41}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

Since \frac{8\times 5\sqrt{41}}{41} and \frac{12\sqrt{41}}{41} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8\times 5\sqrt{41}-12\sqrt{41}}{41}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

Do the multiplications in 8\times 5\sqrt{41}-12\sqrt{41}.

$$\frac{\frac{40\sqrt{41}-12\sqrt{41}}{41}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

Do the calculations in 40\sqrt{41}-12\sqrt{41}.

$$\frac{\frac{28\sqrt{41}}{41}}{8\times \frac{5}{\sqrt{41}}+2\times \frac{4}{\sqrt{41}}}$$

Rationalize the denominator of \frac{5}{\sqrt{41}} by multiplying numerator and denominator by \sqrt{41}.

$$\frac{\frac{28\sqrt{41}}{41}}{8\times \frac{5\sqrt{41}}{\left(\sqrt{41}\right)^{2}}+2\times \frac{4}{\sqrt{41}}}$$

The square of \sqrt{41} is 41.

$$\frac{\frac{28\sqrt{41}}{41}}{8\times \frac{5\sqrt{41}}{41}+2\times \frac{4}{\sqrt{41}}}$$

Express 8\times \frac{5\sqrt{41}}{41} as a single fraction.

$$\frac{\frac{28\sqrt{41}}{41}}{\frac{8\times 5\sqrt{41}}{41}+2\times \frac{4}{\sqrt{41}}}$$

Rationalize the denominator of \frac{4}{\sqrt{41}} by multiplying numerator and denominator by \sqrt{41}.

$$\frac{\frac{28\sqrt{41}}{41}}{\frac{8\times 5\sqrt{41}}{41}+2\times \frac{4\sqrt{41}}{\left(\sqrt{41}\right)^{2}}}$$

The square of \sqrt{41} is 41.

$$\frac{\frac{28\sqrt{41}}{41}}{\frac{8\times 5\sqrt{41}}{41}+2\times \frac{4\sqrt{41}}{41}}$$

Express 2\times \frac{4\sqrt{41}}{41} as a single fraction.

$$\frac{\frac{28\sqrt{41}}{41}}{\frac{8\times 5\sqrt{41}}{41}+\frac{2\times 4\sqrt{41}}{41}}$$

Since \frac{8\times 5\sqrt{41}}{41} and \frac{2\times 4\sqrt{41}}{41} have the same denominator, add them by adding their numerators.

$$\frac{\frac{28\sqrt{41}}{41}}{\frac{8\times 5\sqrt{41}+2\times 4\sqrt{41}}{41}}$$

Do the multiplications in 8\times 5\sqrt{41}+2\times 4\sqrt{41}.

$$\frac{\frac{28\sqrt{41}}{41}}{\frac{40\sqrt{41}+8\sqrt{41}}{41}}$$

Do the calculations in 40\sqrt{41}+8\sqrt{41}.

$$\frac{\frac{28\sqrt{41}}{41}}{\frac{48\sqrt{41}}{41}}$$

Divide \frac{28\sqrt{41}}{41} by \frac{48\sqrt{41}}{41} by multiplying \frac{28\sqrt{41}}{41} by the reciprocal of \frac{48\sqrt{41}}{41}.

$$\frac{28\sqrt{41}\times 41}{41\times 48\sqrt{41}}$$

Cancel out 4\times 41\sqrt{41} in both numerator and denominator.

$$\frac{7}{12}$$

Factor

$$\frac{7}{2 ^ {2} \cdot 3} = 0.5833333333333334$$