Solve 300-frac{700/sqrt{109}}{650+210/sqrt{109}}

$$\frac{ 300- \frac{ 700 }{ \sqrt{ 109 } } }{ 650+ \frac{ 210 }{ \sqrt{ 109 } } }$$

Evaluate

$$\frac{53505-1295\sqrt{109}}{115021}\approx 0.347630459$$

Short Solution Steps

Rationalize the denominator of \frac{700}{\sqrt{109}} by multiplying numerator and denominator by \sqrt{109}.

$$\frac{300-\frac{700\sqrt{109}}{\left(\sqrt{109}\right)^{2}}}{650+\frac{210}{\sqrt{109}}}$$

The square of \sqrt{109} is 109.

$$\frac{300-\frac{700\sqrt{109}}{109}}{650+\frac{210}{\sqrt{109}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 300 times \frac{109}{109}.

$$\frac{\frac{300\times 109}{109}-\frac{700\sqrt{109}}{109}}{650+\frac{210}{\sqrt{109}}}$$

Since \frac{300\times 109}{109} and \frac{700\sqrt{109}}{109} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{300\times 109-700\sqrt{109}}{109}}{650+\frac{210}{\sqrt{109}}}$$

Do the multiplications in 300\times 109-700\sqrt{109}.

$$\frac{\frac{32700-700\sqrt{109}}{109}}{650+\frac{210}{\sqrt{109}}}$$

Rationalize the denominator of \frac{210}{\sqrt{109}} by multiplying numerator and denominator by \sqrt{109}.

$$\frac{\frac{32700-700\sqrt{109}}{109}}{650+\frac{210\sqrt{109}}{\left(\sqrt{109}\right)^{2}}}$$

The square of \sqrt{109} is 109.

$$\frac{\frac{32700-700\sqrt{109}}{109}}{650+\frac{210\sqrt{109}}{109}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 650 times \frac{109}{109}.

$$\frac{\frac{32700-700\sqrt{109}}{109}}{\frac{650\times 109}{109}+\frac{210\sqrt{109}}{109}}$$

Since \frac{650\times 109}{109} and \frac{210\sqrt{109}}{109} have the same denominator, add them by adding their numerators.

$$\frac{\frac{32700-700\sqrt{109}}{109}}{\frac{650\times 109+210\sqrt{109}}{109}}$$

Do the multiplications in 650\times 109+210\sqrt{109}.

$$\frac{\frac{32700-700\sqrt{109}}{109}}{\frac{70850+210\sqrt{109}}{109}}$$

Divide \frac{32700-700\sqrt{109}}{109} by \frac{70850+210\sqrt{109}}{109} by multiplying \frac{32700-700\sqrt{109}}{109} by the reciprocal of \frac{70850+210\sqrt{109}}{109}.

$$\frac{\left(32700-700\sqrt{109}\right)\times 109}{109\left(70850+210\sqrt{109}\right)}$$

Cancel out 109 in both numerator and denominator.

$$\frac{-700\sqrt{109}+32700}{210\sqrt{109}+70850}$$

Rationalize the denominator of \frac{-700\sqrt{109}+32700}{210\sqrt{109}+70850} by multiplying numerator and denominator by 210\sqrt{109}-70850.

$$\frac{\left(-700\sqrt{109}+32700\right)\left(210\sqrt{109}-70850\right)}{\left(210\sqrt{109}+70850\right)\left(210\sqrt{109}-70850\right)}$$

Consider \left(210\sqrt{109}+70850\right)\left(210\sqrt{109}-70850\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(-700\sqrt{109}+32700\right)\left(210\sqrt{109}-70850\right)}{\left(210\sqrt{109}\right)^{2}-70850^{2}}$$

Expand \left(210\sqrt{109}\right)^{2}.

$$\frac{\left(-700\sqrt{109}+32700\right)\left(210\sqrt{109}-70850\right)}{210^{2}\left(\sqrt{109}\right)^{2}-70850^{2}}$$

Calculate 210 to the power of 2 and get 44100.

$$\frac{\left(-700\sqrt{109}+32700\right)\left(210\sqrt{109}-70850\right)}{44100\left(\sqrt{109}\right)^{2}-70850^{2}}$$

The square of \sqrt{109} is 109.

$$\frac{\left(-700\sqrt{109}+32700\right)\left(210\sqrt{109}-70850\right)}{44100\times 109-70850^{2}}$$

Multiply 44100 and 109 to get 4806900.

$$\frac{\left(-700\sqrt{109}+32700\right)\left(210\sqrt{109}-70850\right)}{4806900-70850^{2}}$$

Calculate 70850 to the power of 2 and get 5019722500.

$$\frac{\left(-700\sqrt{109}+32700\right)\left(210\sqrt{109}-70850\right)}{4806900-5019722500}$$

Subtract 5019722500 from 4806900 to get -5014915600.

$$\frac{\left(-700\sqrt{109}+32700\right)\left(210\sqrt{109}-70850\right)}{-5014915600}$$

Apply the distributive property by multiplying each term of -700\sqrt{109}+32700 by each term of 210\sqrt{109}-70850.

$$\frac{-147000\left(\sqrt{109}\right)^{2}[phone]\sqrt{109}+6867000\sqrt{109}-2316795000}{-5014915600}$$

The square of \sqrt{109} is 109.

$$\frac{-147000\times 109+49595000\sqrt{109}+6867000\sqrt{109}-2316795000}{-5014915600}$$

Multiply -147000 and 109 to get -16023000.

$$\frac{-16023000+49595000\sqrt{109}+6867000\sqrt{109}-2316795000}{-5014915600}$$

Combine 49595000\sqrt{109} and 6867000\sqrt{109} to get 56462000\sqrt{109}.

$$\frac{-16023000+56462000\sqrt{109}-2316795000}{-5014915600}$$

Subtract 2316795000 from -16023000 to get -2332818000.

$$\frac{-2332818000+56462000\sqrt{109}}{-5014915600}$$

Factor

$$\frac{5 {(10701 - 259 \sqrt{109})}}{115021} = 0.347630459402725$$