Solve 75.4-56.89/sqrt{frac{11.01^2{87}+6.89^2/111}}

$$\frac{ 75.4-56.89 }{ \sqrt{ \frac{ { 11.01 }^{ 2 } }{ 87 } + \frac{ { 6.89 }^{ 2 } }{ 111 } } }$$

Evaluate

$$\frac{1851\sqrt{188692455774}}{58618346}\approx 13.716715377$$

Solution Steps

Subtract 56.89 from 75.4 to get 18.51.

$$\frac{18.51}{\sqrt{\frac{11.01^{2}}{87}+\frac{6.89^{2}}{111}}}$$

Calculate 11.01 to the power of 2 and get 121.2201.

$$\frac{18.51}{\sqrt{\frac{121.2201}{87}+\frac{6.89^{2}}{111}}}$$

Expand \frac{121.2201}{87} by multiplying both numerator and the denominator by 10000.

$$\frac{18.51}{\sqrt{\frac{1212201}{870000}+\frac{6.89^{2}}{111}}}$$

Reduce the fraction \frac{1212201}{870000} to lowest terms by extracting and canceling out 3.

$$\frac{18.51}{\sqrt{\frac{404067}{290000}+\frac{6.89^{2}}{111}}}$$

Calculate 6.89 to the power of 2 and get 47.4721.

$$\frac{18.51}{\sqrt{\frac{404067}{290000}+\frac{47.4721}{111}}}$$

Expand \frac{47.4721}{111} by multiplying both numerator and the denominator by 10000.

$$\frac{18.51}{\sqrt{\frac{404067}{290000}+\frac{474721}{1110000}}}$$

Least common multiple of 290000 and 1110000 is 32190000. Convert \frac{404067}{290000} and \frac{474721}{1110000} to fractions with denominator 32190000.

$$\frac{18.51}{\sqrt{\frac{44851437}{32190000}+\frac{13766909}{32190000}}}$$

Since \frac{44851437}{32190000} and \frac{13766909}{32190000} have the same denominator, add them by adding their numerators.

$$\frac{18.51}{\sqrt{\frac{44851437+13766909}{32190000}}}$$

Add 44851437 and 13766909 to get 58618346.

$$\frac{18.51}{\sqrt{\frac{58618346}{32190000}}}$$

Reduce the fraction \frac{58618346}{32190000} to lowest terms by extracting and canceling out 2.

$$\frac{18.51}{\sqrt{\frac{29309173}{16095000}}}$$

Rewrite the square root of the division \sqrt{\frac{29309173}{16095000}} as the division of square roots \frac{\sqrt{29309173}}{\sqrt{16095000}}.

$$\frac{18.51}{\frac{\sqrt{29309173}}{\sqrt{16095000}}}$$

Factor 16095000=50^{2}\times 6438. Rewrite the square root of the product \sqrt{50^{2}\times 6438} as the product of square roots \sqrt{50^{2}}\sqrt{6438}. Take the square root of 50^{2}.

$$\frac{18.51}{\frac{\sqrt{29309173}}{50\sqrt{6438}}}$$

Rationalize the denominator of \frac{\sqrt{29309173}}{50\sqrt{6438}} by multiplying numerator and denominator by \sqrt{6438}.

$$\frac{18.51}{\frac{\sqrt{29309173}\sqrt{6438}}{50\left(\sqrt{6438}\right)^{2}}}$$

The square of \sqrt{6438} is 6438.

$$\frac{18.51}{\frac{\sqrt{29309173}\sqrt{6438}}{50\times 6438}}$$

To multiply \sqrt{29309173} and \sqrt{6438}, multiply the numbers under the square root.

$$\frac{18.51}{\frac{\sqrt{188692455774}}{50\times 6438}}$$

Multiply 50 and 6438 to get 321900.

$$\frac{18.51}{\frac{\sqrt{188692455774}}{321900}}$$

Divide 18.51 by \frac{\sqrt{188692455774}}{321900} by multiplying 18.51 by the reciprocal of \frac{\sqrt{188692455774}}{321900}.

$$\frac{18.51\times 321900}{\sqrt{188692455774}}$$

Rationalize the denominator of \frac{18.51\times 321900}{\sqrt{188692455774}} by multiplying numerator and denominator by \sqrt{188692455774}.

$$\frac{18.51\times 321900\sqrt{188692455774}}{\left(\sqrt{188692455774}\right)^{2}}$$

The square of \sqrt{188692455774} is 188692455774.

$$\frac{18.51\times 321900\sqrt{188692455774}}{188692455774}$$

Multiply 18.51 and 321900 to get 5958369.

$$\frac{5958369\sqrt{188692455774}}{188692455774}$$

Divide 5958369\sqrt{188692455774} by 188692455774 to get \frac{1851}{58618346}\sqrt{188692455774}.

$$\frac{1851}{58618346}\sqrt{188692455774}$$