Solve 0.78933-0.81391-0/sqrt{frac{0.00436^2{23}+0.00757^2/23}}

$$\frac{ 0.78933-0.81391-0 }{ \sqrt{ \frac{ { 0.00436 }^{ 2 } }{ 23 } + \frac{ { 0.00757 }^{ 2 } }{ 23 } } }$$

Evaluate

$$-\frac{2458\sqrt{17552335}}{763145}\approx -13.494048445$$

Solution Steps

Subtract 0.81391 from 0.78933 to get -0.02458.

$$\frac{-0.02458-0}{\sqrt{\frac{0.00436^{2}}{23}+\frac{0.00757^{2}}{23}}}$$

Subtract 0 from -0.02458 to get -0.02458.

$$\frac{-0.02458}{\sqrt{\frac{0.00436^{2}}{23}+\frac{0.00757^{2}}{23}}}$$

Calculate 0.00436 to the power of 2 and get 0.0000190096.

$$\frac{-0.02458}{\sqrt{\frac{0.0000190096}{23}+\frac{0.00757^{2}}{23}}}$$

Expand \frac{0.0000190096}{23} by multiplying both numerator and the denominator by 10000000000.

$$\frac{-0.02458}{\sqrt{\frac{190096}{230000000000}+\frac{0.00757^{2}}{23}}}$$

Reduce the fraction \frac{190096}{230000000000} to lowest terms by extracting and canceling out 16.

$$\frac{-0.02458}{\sqrt{\frac{11881}{14375000000}+\frac{0.00757^{2}}{23}}}$$

Calculate 0.00757 to the power of 2 and get 0.0000573049.

$$\frac{-0.02458}{\sqrt{\frac{11881}{14375000000}+\frac{0.0000573049}{23}}}$$

Expand \frac{0.0000573049}{23} by multiplying both numerator and the denominator by 10000000000.

$$\frac{-0.02458}{\sqrt{\frac{11881}{14375000000}+\frac{573049}{230000000000}}}$$

Least common multiple of 14375000000 and 230000000000 is 230000000000. Convert \frac{11881}{14375000000} and \frac{573049}{230000000000} to fractions with denominator 230000000000.

$$\frac{-0.02458}{\sqrt{\frac{190096}{230000000000}+\frac{573049}{230000000000}}}$$

Since \frac{190096}{230000000000} and \frac{573049}{230000000000} have the same denominator, add them by adding their numerators.

$$\frac{-0.02458}{\sqrt{\frac{190096+573049}{230000000000}}}$$

Add 190096 and 573049 to get 763145.

$$\frac{-0.02458}{\sqrt{\frac{763145}{230000000000}}}$$

Reduce the fraction \frac{763145}{230000000000} to lowest terms by extracting and canceling out 5.

$$\frac{-0.02458}{\sqrt{\frac{152629}{46000000000}}}$$

Rewrite the square root of the division \sqrt{\frac{152629}{46000000000}} as the division of square roots \frac{\sqrt{152629}}{\sqrt{46000000000}}.

$$\frac{-0.02458}{\frac{\sqrt{152629}}{\sqrt{46000000000}}}$$

Factor 46000000000=20000^{2}\times 115. Rewrite the square root of the product \sqrt{20000^{2}\times 115} as the product of square roots \sqrt{20000^{2}}\sqrt{115}. Take the square root of 20000^{2}.

$$\frac{-0.02458}{\frac{\sqrt{152629}}{20000\sqrt{115}}}$$

Rationalize the denominator of \frac{\sqrt{152629}}{20000\sqrt{115}} by multiplying numerator and denominator by \sqrt{115}.

$$\frac{-0.02458}{\frac{\sqrt{152629}\sqrt{115}}{20000\left(\sqrt{115}\right)^{2}}}$$

The square of \sqrt{115} is 115.

$$\frac{-0.02458}{\frac{\sqrt{152629}\sqrt{115}}{20000\times 115}}$$

To multiply \sqrt{152629} and \sqrt{115}, multiply the numbers under the square root.

$$\frac{-0.02458}{\frac{\sqrt{17552335}}{20000\times 115}}$$

Multiply 20000 and 115 to get 2300000.

$$\frac{-0.02458}{\frac{\sqrt{17552335}}{2300000}}$$

Divide -0.02458 by \frac{\sqrt{17552335}}{2300000} by multiplying -0.02458 by the reciprocal of \frac{\sqrt{17552335}}{2300000}.

$$\frac{-0.02458\times 2300000}{\sqrt{17552335}}$$

Rationalize the denominator of \frac{-0.02458\times 2300000}{\sqrt{17552335}} by multiplying numerator and denominator by \sqrt{17552335}.

$$\frac{-0.02458\times 2300000\sqrt{17552335}}{\left(\sqrt{17552335}\right)^{2}}$$

The square of \sqrt{17552335} is 17552335.

$$\frac{-0.02458\times 2300000\sqrt{17552335}}{17552335}$$

Multiply -0.02458 and 2300000 to get -56534.

$$\frac{-56534\sqrt{17552335}}{17552335}$$

Divide -56534\sqrt{17552335} by 17552335 to get -\frac{2458}{763145}\sqrt{17552335}.

$$-\frac{2458}{763145}\sqrt{17552335}$$