Solve {left(300-700/sqrt{109}right)}^2+{left(650+210/sqrt{109}right)}^2

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

Evaluate

$$-\frac{147000\sqrt{109}}{109}+517400\approx 503319.953607249$$

Short Solution Steps

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

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

The square of \sqrt{109} is 109.

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

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

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

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

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

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

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

To raise \frac{32700-700\sqrt{109}}{109} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

The square of \sqrt{109} is 109.

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

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

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

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

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

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

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

To raise \frac{70850+210\sqrt{109}}{109} to a power, raise both numerator and denominator to the power and then divide.

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

Since \frac{\left(32700-700\sqrt{109}\right)^{2}}{109^{2}} and \frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}} have the same denominator, add them by adding their numerators.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(32700-700\sqrt{109}\right)^{2}.

$$\frac{1069290000-45780000\sqrt{109}+490000\left(\sqrt{109}\right)^{2}}{109^{2}}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

The square of \sqrt{109} is 109.

$$\frac{1069290000-45780000\sqrt{109}+490000\times 109}{109^{2}}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

Multiply 490000 and 109 to get 53410000.

$$\frac{1069290000-45780000\sqrt{109}[phone]}{109^{2}}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

Add 1069290000 and 53410000 to get 1122700000.

$$\frac{1122700000-45780000\sqrt{109}}{109^{2}}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

Calculate 109 to the power of 2 and get 11881.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(70850+210\sqrt{109}\right)^{2}.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5019722500+29757000\sqrt{109}+44100\left(\sqrt{109}\right)^{2}}{109^{2}}$$

The square of \sqrt{109} is 109.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5019722500+29757000\sqrt{109}+44100\times 109}{109^{2}}$$

Multiply 44100 and 109 to get 4806900.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5019722500+29757000\sqrt{109}+4806900}{109^{2}}$$

Add 5019722500 and 4806900 to get 5024529400.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5024529400+29757000\sqrt{109}}{109^{2}}$$

Calculate 109 to the power of 2 and get 11881.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5024529400+29757000\sqrt{109}}{11881}$$

Since \frac{1122700000-45780000\sqrt{109}}{11881} and \frac{5024529400+29757000\sqrt{109}}{11881} have the same denominator, add them by adding their numerators.

$$\frac{1122700000-45780000\sqrt{109}[phone]+29757000\sqrt{109}}{11881}$$

Do the calculations in 1122700000-45780000\sqrt{109}[phone]+29757000\sqrt{109}.

$$\frac{6147229400-16023000\sqrt{109}}{11881}$$

Expand

$$-\frac{147000 \sqrt{109}}{109} + 517400 = 503319.953607249$$

Short Solution Steps

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

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

The square of \sqrt{109} is 109.

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

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

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

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

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

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

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

To raise \frac{32700-700\sqrt{109}}{109} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

The square of \sqrt{109} is 109.

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

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

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

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

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

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

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

To raise \frac{70850+210\sqrt{109}}{109} to a power, raise both numerator and denominator to the power and then divide.

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

Since \frac{\left(32700-700\sqrt{109}\right)^{2}}{109^{2}} and \frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}} have the same denominator, add them by adding their numerators.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(32700-700\sqrt{109}\right)^{2}.

$$\frac{1069290000-45780000\sqrt{109}+490000\left(\sqrt{109}\right)^{2}}{109^{2}}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

The square of \sqrt{109} is 109.

$$\frac{1069290000-45780000\sqrt{109}+490000\times 109}{109^{2}}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

Multiply 490000 and 109 to get 53410000.

$$\frac{1069290000-45780000\sqrt{109}[phone]}{109^{2}}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

Add 1069290000 and 53410000 to get 1122700000.

$$\frac{1122700000-45780000\sqrt{109}}{109^{2}}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

Calculate 109 to the power of 2 and get 11881.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{\left(70850+210\sqrt{109}\right)^{2}}{109^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(70850+210\sqrt{109}\right)^{2}.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5019722500+29757000\sqrt{109}+44100\left(\sqrt{109}\right)^{2}}{109^{2}}$$

The square of \sqrt{109} is 109.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5019722500+29757000\sqrt{109}+44100\times 109}{109^{2}}$$

Multiply 44100 and 109 to get 4806900.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5019722500+29757000\sqrt{109}+4806900}{109^{2}}$$

Add 5019722500 and 4806900 to get 5024529400.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5024529400+29757000\sqrt{109}}{109^{2}}$$

Calculate 109 to the power of 2 and get 11881.

$$\frac{1122700000-45780000\sqrt{109}}{11881}+\frac{5024529400+29757000\sqrt{109}}{11881}$$

Since \frac{1122700000-45780000\sqrt{109}}{11881} and \frac{5024529400+29757000\sqrt{109}}{11881} have the same denominator, add them by adding their numerators.

$$\frac{1122700000-45780000\sqrt{109}[phone]+29757000\sqrt{109}}{11881}$$

Do the calculations in 1122700000-45780000\sqrt{109}[phone]+29757000\sqrt{109}.

$$\frac{6147229400-16023000\sqrt{109}}{11881}$$