Solve 39472-24751-10000/sqrt{frac{31592^2{137}+12760^2/431}}

$$\frac{ 39472-24751-10000 }{ \sqrt{ \frac{ { 31592 }^{ 2 } }{ 137 } + \frac{ { 12760 }^{ 2 } }{ 431 } } }$$

Evaluate

$$\frac{4721\sqrt{862501536598}}{2570837984}\approx 1.705450942$$

Solution Steps

Subtract 24751 from 39472 to get 14721.

$$\frac{14721-10000}{\sqrt{\frac{31592^{2}}{137}+\frac{12760^{2}}{431}}}$$

Subtract 10000 from 14721 to get 4721.

$$\frac{4721}{\sqrt{\frac{31592^{2}}{137}+\frac{12760^{2}}{431}}}$$

Calculate 31592 to the power of 2 and get 998054464.

$$\frac{4721}{\sqrt{\frac{998054464}{137}+\frac{12760^{2}}{431}}}$$

Calculate 12760 to the power of 2 and get 162817600.

$$\frac{4721}{\sqrt{\frac{998054464}{137}+\frac{162817600}{431}}}$$

Least common multiple of 137 and 431 is 59047. Convert \frac{998054464}{137} and \frac{162817600}{431} to fractions with denominator 59047.

$$\frac{4721}{\sqrt{\frac{430161473984}{59047}+\frac{22306011200}{59047}}}$$

Since \frac{430161473984}{59047} and \frac{22306011200}{59047} have the same denominator, add them by adding their numerators.

$$\frac{4721}{\sqrt{\frac{430161473984+22306011200}{59047}}}$$

Add 430161473984 and 22306011200 to get 452467485184.

$$\frac{4721}{\sqrt{\frac{452467485184}{59047}}}$$

Rewrite the square root of the division \sqrt{\frac{452467485184}{59047}} as the division of square roots \frac{\sqrt{452467485184}}{\sqrt{59047}}.

$$\frac{4721}{\frac{\sqrt{452467485184}}{\sqrt{59047}}}$$

Factor 452467485184=176^{2}\times 14607034. Rewrite the square root of the product \sqrt{176^{2}\times 14607034} as the product of square roots \sqrt{176^{2}}\sqrt{14607034}. Take the square root of 176^{2}.

$$\frac{4721}{\frac{176\sqrt{14607034}}{\sqrt{59047}}}$$

Rationalize the denominator of \frac{176\sqrt{14607034}}{\sqrt{59047}} by multiplying numerator and denominator by \sqrt{59047}.

$$\frac{4721}{\frac{176\sqrt{14607034}\sqrt{59047}}{\left(\sqrt{59047}\right)^{2}}}$$

The square of \sqrt{59047} is 59047.

$$\frac{4721}{\frac{176\sqrt{14607034}\sqrt{59047}}{59047}}$$

To multiply \sqrt{14607034} and \sqrt{59047}, multiply the numbers under the square root.

$$\frac{4721}{\frac{176\sqrt{862501536598}}{59047}}$$

Divide 4721 by \frac{176\sqrt{862501536598}}{59047} by multiplying 4721 by the reciprocal of \frac{176\sqrt{862501536598}}{59047}.

$$\frac{4721\times 59047}{176\sqrt{862501536598}}$$

Rationalize the denominator of \frac{4721\times 59047}{176\sqrt{862501536598}} by multiplying numerator and denominator by \sqrt{862501536598}.

$$\frac{4721\times 59047\sqrt{862501536598}}{176\left(\sqrt{862501536598}\right)^{2}}$$

The square of \sqrt{862501536598} is 862501536598.

$$\frac{4721\times 59047\sqrt{862501536598}}{176\times 862501536598}$$

Multiply 4721 and 59047 to get 278760887.

$$\frac{278760887\sqrt{862501536598}}{176\times 862501536598}$$

Multiply 176 and 862501536598 to get 151800270441248.

$$\frac{278760887\sqrt{862501536598}}{151800270441248}$$

Divide 278760887\sqrt{862501536598} by 151800270441248 to get \frac{4721}{2570837984}\sqrt{862501536598}.

$$\frac{4721}{2570837984}\sqrt{862501536598}$$