Solve 0.5693-0.4738/sqrt{frac{1.0069{165}+1.3564/165}}

$$\frac { 0.5693 - 0.4738 } { \sqrt { \frac { 1.0069 } { 165 } + \frac { 1.3564 } { 165 } } }$$

Evaluate

$$\frac{191\sqrt{3899445}}{472660}\approx 0.797968809$$

Solution Steps

Subtract 0.4738 from 0.5693 to get 0.0955.

$$\frac{0.0955}{\sqrt{\frac{1.0069}{165}+\frac{1.3564}{165}}}$$

Expand \frac{1.0069}{165} by multiplying both numerator and the denominator by 10000.

$$\frac{0.0955}{\sqrt{\frac{10069}{1650000}+\frac{1.3564}{165}}}$$

Expand \frac{1.3564}{165} by multiplying both numerator and the denominator by 10000.

$$\frac{0.0955}{\sqrt{\frac{10069}{1650000}+\frac{13564}{1650000}}}$$

Since \frac{10069}{1650000} and \frac{13564}{1650000} have the same denominator, add them by adding their numerators.

$$\frac{0.0955}{\sqrt{\frac{10069+13564}{1650000}}}$$

Add 10069 and 13564 to get 23633.

$$\frac{0.0955}{\sqrt{\frac{23633}{1650000}}}$$

Rewrite the square root of the division \sqrt{\frac{23633}{1650000}} as the division of square roots \frac{\sqrt{23633}}{\sqrt{1650000}}.

$$\frac{0.0955}{\frac{\sqrt{23633}}{\sqrt{1650000}}}$$

Factor 1650000=100^{2}\times 165. Rewrite the square root of the product \sqrt{100^{2}\times 165} as the product of square roots \sqrt{100^{2}}\sqrt{165}. Take the square root of 100^{2}.

$$\frac{0.0955}{\frac{\sqrt{23633}}{100\sqrt{165}}}$$

Rationalize the denominator of \frac{\sqrt{23633}}{100\sqrt{165}} by multiplying numerator and denominator by \sqrt{165}.

$$\frac{0.0955}{\frac{\sqrt{23633}\sqrt{165}}{100\left(\sqrt{165}\right)^{2}}}$$

The square of \sqrt{165} is 165.

$$\frac{0.0955}{\frac{\sqrt{23633}\sqrt{165}}{100\times 165}}$$

To multiply \sqrt{23633} and \sqrt{165}, multiply the numbers under the square root.

$$\frac{0.0955}{\frac{\sqrt{3899445}}{100\times 165}}$$

Multiply 100 and 165 to get 16500.

$$\frac{0.0955}{\frac{\sqrt{3899445}}{16500}}$$

Divide 0.0955 by \frac{\sqrt{3899445}}{16500} by multiplying 0.0955 by the reciprocal of \frac{\sqrt{3899445}}{16500}.

$$\frac{0.0955\times 16500}{\sqrt{3899445}}$$

Rationalize the denominator of \frac{0.0955\times 16500}{\sqrt{3899445}} by multiplying numerator and denominator by \sqrt{3899445}.

$$\frac{0.0955\times 16500\sqrt{3899445}}{\left(\sqrt{3899445}\right)^{2}}$$

The square of \sqrt{3899445} is 3899445.

$$\frac{0.0955\times 16500\sqrt{3899445}}{3899445}$$

Multiply 0.0955 and 16500 to get 1575.75.

$$\frac{1575.75\sqrt{3899445}}{3899445}$$

Divide 1575.75\sqrt{3899445} by 3899445 to get \frac{191}{472660}\sqrt{3899445}.

$$\frac{191}{472660}\sqrt{3899445}$$