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

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

Evaluate

$$\frac{738\sqrt{3}+4665}{169}\approx 35.16718045$$

Short Solution Steps

Multiply \sqrt{\left(\frac{-12\sqrt{3}-9}{13}-2\sqrt{3}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}} and \sqrt{\left(\frac{-12\sqrt{3}-9}{13}-2\sqrt{3}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}} to get \left(\sqrt{\left(\frac{-12\sqrt{3}-9}{13}-2\sqrt{3}\right)^{2}+\left(\frac{3+9\sqrt{3}}{13}\right)^{2}}\right)^{2}.

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

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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

Calculate \sqrt{\frac{\left(-38\sqrt{3}-9\right)^{2}+\left(3+9\sqrt{3}\right)^{2}}{13^{2}}} to the power of 2 and get \frac{\left(-38\sqrt{3}-9\right)^{2}+\left(3+9\sqrt{3}\right)^{2}}{13^{2}}.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 1444 and 3 to get 4332.

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

Add 4332 and 81 to get 4413.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 81 and 3 to get 243.

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

Add 9 and 243 to get 252.

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

Add 4413 and 252 to get 4665.

$$\frac{4665+684\sqrt{3}+54\sqrt{3}}{13^{2}}$$

Combine 684\sqrt{3} and 54\sqrt{3} to get 738\sqrt{3}.

$$\frac{4665+738\sqrt{3}}{13^{2}}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{4665+738\sqrt{3}}{169}$$

Factor

$$\frac{3 {(246 \sqrt{3} + 1555)}}{169} = 35.167180449620304$$