Solve 400/850-0.55/sqrt{frac{0.55*0.45{850}}}

$$\frac{ \frac{ 400 }{ 850 } -0.55 }{ \sqrt{ \frac{ 0.55 \times 0.45 }{ 850 } } }$$

Evaluate

$$-\frac{45\sqrt{374}}{187}\approx -4.65378921$$

Solution Steps

Reduce the fraction \frac{400}{850} to lowest terms by extracting and canceling out 50.

$$\frac{\frac{8}{17}-0.55}{\sqrt{\frac{0.55\times 0.45}{850}}}$$

Convert decimal number 0.55 to fraction \frac{55}{100}. Reduce the fraction \frac{55}{100} to lowest terms by extracting and canceling out 5.

$$\frac{\frac{8}{17}-\frac{11}{20}}{\sqrt{\frac{0.55\times 0.45}{850}}}$$

Least common multiple of 17 and 20 is 340. Convert \frac{8}{17} and \frac{11}{20} to fractions with denominator 340.

$$\frac{\frac{160}{340}-\frac{187}{340}}{\sqrt{\frac{0.55\times 0.45}{850}}}$$

Since \frac{160}{340} and \frac{187}{340} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{160-187}{340}}{\sqrt{\frac{0.55\times 0.45}{850}}}$$

Subtract 187 from 160 to get -27.

$$\frac{-\frac{27}{340}}{\sqrt{\frac{0.55\times 0.45}{850}}}$$

Multiply 0.55 and 0.45 to get 0.2475.

$$\frac{-\frac{27}{340}}{\sqrt{\frac{0.2475}{850}}}$$

Expand \frac{0.2475}{850} by multiplying both numerator and the denominator by 10000.

$$\frac{-\frac{27}{340}}{\sqrt{\frac{2475}{8500000}}}$$

Reduce the fraction \frac{2475}{8500000} to lowest terms by extracting and canceling out 25.

$$\frac{-\frac{27}{340}}{\sqrt{\frac{99}{340000}}}$$

Rewrite the square root of the division \sqrt{\frac{99}{340000}} as the division of square roots \frac{\sqrt{99}}{\sqrt{340000}}.

$$\frac{-\frac{27}{340}}{\frac{\sqrt{99}}{\sqrt{340000}}}$$

Factor 99=3^{2}\times 11. Rewrite the square root of the product \sqrt{3^{2}\times 11} as the product of square roots \sqrt{3^{2}}\sqrt{11}. Take the square root of 3^{2}.

$$\frac{-\frac{27}{340}}{\frac{3\sqrt{11}}{\sqrt{340000}}}$$

Factor 340000=100^{2}\times 34. Rewrite the square root of the product \sqrt{100^{2}\times 34} as the product of square roots \sqrt{100^{2}}\sqrt{34}. Take the square root of 100^{2}.

$$\frac{-\frac{27}{340}}{\frac{3\sqrt{11}}{100\sqrt{34}}}$$

Rationalize the denominator of \frac{3\sqrt{11}}{100\sqrt{34}} by multiplying numerator and denominator by \sqrt{34}.

$$\frac{-\frac{27}{340}}{\frac{3\sqrt{11}\sqrt{34}}{100\left(\sqrt{34}\right)^{2}}}$$

The square of \sqrt{34} is 34.

$$\frac{-\frac{27}{340}}{\frac{3\sqrt{11}\sqrt{34}}{100\times 34}}$$

To multiply \sqrt{11} and \sqrt{34}, multiply the numbers under the square root.

$$\frac{-\frac{27}{340}}{\frac{3\sqrt{374}}{100\times 34}}$$

Multiply 100 and 34 to get 3400.

$$\frac{-\frac{27}{340}}{\frac{3\sqrt{374}}{3400}}$$

Divide -\frac{27}{340} by \frac{3\sqrt{374}}{3400} by multiplying -\frac{27}{340} by the reciprocal of \frac{3\sqrt{374}}{3400}.

$$\frac{-27\times 3400}{340\times 3\sqrt{374}}$$

Cancel out 3\times 340 in both numerator and denominator.

$$\frac{-9\times 10}{\sqrt{374}}$$

Rationalize the denominator of \frac{-9\times 10}{\sqrt{374}} by multiplying numerator and denominator by \sqrt{374}.

$$\frac{-9\times 10\sqrt{374}}{\left(\sqrt{374}\right)^{2}}$$

The square of \sqrt{374} is 374.

$$\frac{-9\times 10\sqrt{374}}{374}$$

Multiply -9 and 10 to get -90.

$$\frac{-90\sqrt{374}}{374}$$

Divide -90\sqrt{374} by 374 to get -\frac{45}{187}\sqrt{374}.

$$-\frac{45}{187}\sqrt{374}$$