Solve (frac{-12sqrt{3}+9}{13}-2sqrt{3})^2+(frac{3+9sqrt{3}}{13})^2

$$( \frac { - 12 \sqrt { 3 } + 9 } { 13 } - 2 \sqrt { 3 } ) ^ { 2 } + ( \frac { 3 + 9 \sqrt { 3 } } { 13 } ) ^ { 2 }$$

Evaluate

$$\frac{4665-630\sqrt{3}}{169}\approx 21.146792848$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -2\sqrt{3} times \frac{13}{13}.

$$\left(\frac{-12\sqrt{3}+9}{13}+\frac{13\left(-2\right)\sqrt{3}}{13}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

Since \frac{-12\sqrt{3}+9}{13} and \frac{13\left(-2\right)\sqrt{3}}{13} have the same denominator, add them by adding their numerators.

$$\left(\frac{-12\sqrt{3}+9+13\left(-2\right)\sqrt{3}}{13}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

Do the multiplications in -12\sqrt{3}+9+13\left(-2\right)\sqrt{3}.

$$\left(\frac{-12\sqrt{3}+9-26\sqrt{3}}{13}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

Do the calculations in -12\sqrt{3}+9-26\sqrt{3}.

$$\left(\frac{-38\sqrt{3}+9}{13}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

To raise \frac{-38\sqrt{3}+9}{13} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-38\sqrt{3}+9\right)^{2}}{13^{2}}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

To raise \frac{3+9\sqrt{3}}{13} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-38\sqrt{3}+9\right)^{2}}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Since \frac{\left(-38\sqrt{3}+9\right)^{2}}{13^{2}} and \frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-38\sqrt{3}+9\right)^{2}+\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-38\sqrt{3}+9\right)^{2}.

$$\frac{1444\left(\sqrt{3}\right)^{2}-684\sqrt{3}+81}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{1444\times 3-684\sqrt{3}+81}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Multiply 1444 and 3 to get 4332.

$$\frac{4332-684\sqrt{3}+81}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Add 4332 and 81 to get 4413.

$$\frac{4413-684\sqrt{3}}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{4413-684\sqrt{3}}{169}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+9\sqrt{3}\right)^{2}.

$$\frac{4413-684\sqrt{3}}{169}+\frac{9+54\sqrt{3}+81\left(\sqrt{3}\right)^{2}}{13^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{4413-684\sqrt{3}}{169}+\frac{9+54\sqrt{3}+81\times 3}{13^{2}}$$

Multiply 81 and 3 to get 243.

$$\frac{4413-684\sqrt{3}}{169}+\frac{9+54\sqrt{3}+243}{13^{2}}$$

Add 9 and 243 to get 252.

$$\frac{4413-684\sqrt{3}}{169}+\frac{252+54\sqrt{3}}{13^{2}}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{4413-684\sqrt{3}}{169}+\frac{252+54\sqrt{3}}{169}$$

Since \frac{4413-684\sqrt{3}}{169} and \frac{252+54\sqrt{3}}{169} have the same denominator, add them by adding their numerators.

$$\frac{4413-684\sqrt{3}+252+54\sqrt{3}}{169}$$

Do the calculations in 4413-684\sqrt{3}+252+54\sqrt{3}.

$$\frac{4665-630\sqrt{3}}{169}$$

Expand

$$\frac{4665 - 630 \sqrt{3}}{169} = 21.146792847524303$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -2\sqrt{3} times \frac{13}{13}.

$$\left(\frac{-12\sqrt{3}+9}{13}+\frac{13\left(-2\right)\sqrt{3}}{13}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

Since \frac{-12\sqrt{3}+9}{13} and \frac{13\left(-2\right)\sqrt{3}}{13} have the same denominator, add them by adding their numerators.

$$\left(\frac{-12\sqrt{3}+9+13\left(-2\right)\sqrt{3}}{13}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

Do the multiplications in -12\sqrt{3}+9+13\left(-2\right)\sqrt{3}.

$$\left(\frac{-12\sqrt{3}+9-26\sqrt{3}}{13}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

Do the calculations in -12\sqrt{3}+9-26\sqrt{3}.

$$\left(\frac{-38\sqrt{3}+9}{13}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

To raise \frac{-38\sqrt{3}+9}{13} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-38\sqrt{3}+9\right)^{2}}{13^{2}}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}$$

To raise \frac{3+9\sqrt{3}}{13} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-38\sqrt{3}+9\right)^{2}}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Since \frac{\left(-38\sqrt{3}+9\right)^{2}}{13^{2}} and \frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-38\sqrt{3}+9\right)^{2}+\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-38\sqrt{3}+9\right)^{2}.

$$\frac{1444\left(\sqrt{3}\right)^{2}-684\sqrt{3}+81}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{1444\times 3-684\sqrt{3}+81}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Multiply 1444 and 3 to get 4332.

$$\frac{4332-684\sqrt{3}+81}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Add 4332 and 81 to get 4413.

$$\frac{4413-684\sqrt{3}}{13^{2}}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{4413-684\sqrt{3}}{169}+\frac{\left(3+9\sqrt{3}\right)^{2}}{13^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+9\sqrt{3}\right)^{2}.

$$\frac{4413-684\sqrt{3}}{169}+\frac{9+54\sqrt{3}+81\left(\sqrt{3}\right)^{2}}{13^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{4413-684\sqrt{3}}{169}+\frac{9+54\sqrt{3}+81\times 3}{13^{2}}$$

Multiply 81 and 3 to get 243.

$$\frac{4413-684\sqrt{3}}{169}+\frac{9+54\sqrt{3}+243}{13^{2}}$$

Add 9 and 243 to get 252.

$$\frac{4413-684\sqrt{3}}{169}+\frac{252+54\sqrt{3}}{13^{2}}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{4413-684\sqrt{3}}{169}+\frac{252+54\sqrt{3}}{169}$$

Since \frac{4413-684\sqrt{3}}{169} and \frac{252+54\sqrt{3}}{169} have the same denominator, add them by adding their numerators.

$$\frac{4413-684\sqrt{3}+252+54\sqrt{3}}{169}$$

Do the calculations in 4413-684\sqrt{3}+252+54\sqrt{3}.

$$\frac{4665-630\sqrt{3}}{169}$$