Solve sqrt{{left(3-2sqrt{6}right)}^2+{left(5sqrt{6}right)}^2+{left(-3-2sqrt{6}right)}^2}

$$\sqrt{ { \left(3-2 \sqrt{ 6 } \right) }^{ 2 } + { \left(5 \sqrt{ 6 } \right) }^{ 2 } + { \left(-3-2 \sqrt{ 6 } \right) }^{ 2 } }$$

Evaluate

$$6\sqrt{6}\approx 14.696938457$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-2\sqrt{6}\right)^{2}.

$$\sqrt{9-12\sqrt{6}+4\left(\sqrt{6}\right)^{2}+\left(5\sqrt{6}\right)^{2}+\left(-3-2\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\sqrt{9-12\sqrt{6}+4\times 6+\left(5\sqrt{6}\right)^{2}+\left(-3-2\sqrt{6}\right)^{2}}$$

Multiply 4 and 6 to get 24.

$$\sqrt{9-12\sqrt{6}+24+\left(5\sqrt{6}\right)^{2}+\left(-3-2\sqrt{6}\right)^{2}}$$

Add 9 and 24 to get 33.

$$\sqrt{33-12\sqrt{6}+\left(5\sqrt{6}\right)^{2}+\left(-3-2\sqrt{6}\right)^{2}}$$

Expand \left(5\sqrt{6}\right)^{2}.

$$\sqrt{33-12\sqrt{6}+5^{2}\left(\sqrt{6}\right)^{2}+\left(-3-2\sqrt{6}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{33-12\sqrt{6}+25\left(\sqrt{6}\right)^{2}+\left(-3-2\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\sqrt{33-12\sqrt{6}+25\times 6+\left(-3-2\sqrt{6}\right)^{2}}$$

Multiply 25 and 6 to get 150.

$$\sqrt{33-12\sqrt{6}+150+\left(-3-2\sqrt{6}\right)^{2}}$$

Add 33 and 150 to get 183.

$$\sqrt{183-12\sqrt{6}+\left(-3-2\sqrt{6}\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-3-2\sqrt{6}\right)^{2}.

$$\sqrt{183-12\sqrt{6}+9+12\sqrt{6}+4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\sqrt{183-12\sqrt{6}+9+12\sqrt{6}+4\times 6}$$

Multiply 4 and 6 to get 24.

$$\sqrt{183-12\sqrt{6}+9+12\sqrt{6}+24}$$

Add 9 and 24 to get 33.

$$\sqrt{183-12\sqrt{6}+33+12\sqrt{6}}$$

Add 183 and 33 to get 216.

$$\sqrt{216-12\sqrt{6}+12\sqrt{6}}$$

Combine -12\sqrt{6} and 12\sqrt{6} to get 0.

$$\sqrt{216}$$

Factor 216=6^{2}\times 6. Rewrite the square root of the product \sqrt{6^{2}\times 6} as the product of square roots \sqrt{6^{2}}\sqrt{6}. Take the square root of 6^{2}.

$$6\sqrt{6}$$