Solve (2sqrt{3}-sqrt{6})^2+(sqrt{54}+2sqrt{6})divsqrt{3}

$$( 2 \sqrt { 3 } - \sqrt { 6 } ) ^ { 2 } + ( \sqrt { 54 } + 2 \sqrt { 6 } ) \div \sqrt { 3 }$$

Evaluate

$$18-7\sqrt{2}\approx 8.100505063$$

Solution Steps

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

$$4\left(\sqrt{3}\right)^{2}-4\sqrt{3}\sqrt{6}+\left(\sqrt{6}\right)^{2}+\frac{\sqrt{54}+2\sqrt{6}}{\sqrt{3}}$$

The square of \sqrt{3} is 3.

$$4\times 3-4\sqrt{3}\sqrt{6}+\left(\sqrt{6}\right)^{2}+\frac{\sqrt{54}+2\sqrt{6}}{\sqrt{3}}$$

Multiply 4 and 3 to get 12.

$$12-4\sqrt{3}\sqrt{6}+\left(\sqrt{6}\right)^{2}+\frac{\sqrt{54}+2\sqrt{6}}{\sqrt{3}}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$12-4\sqrt{3}\sqrt{3}\sqrt{2}+\left(\sqrt{6}\right)^{2}+\frac{\sqrt{54}+2\sqrt{6}}{\sqrt{3}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$12-4\times 3\sqrt{2}+\left(\sqrt{6}\right)^{2}+\frac{\sqrt{54}+2\sqrt{6}}{\sqrt{3}}$$

Multiply -4 and 3 to get -12.

$$12-12\sqrt{2}+\left(\sqrt{6}\right)^{2}+\frac{\sqrt{54}+2\sqrt{6}}{\sqrt{3}}$$

The square of \sqrt{6} is 6.

$$12-12\sqrt{2}+6+\frac{\sqrt{54}+2\sqrt{6}}{\sqrt{3}}$$

Add 12 and 6 to get 18.

$$18-12\sqrt{2}+\frac{\sqrt{54}+2\sqrt{6}}{\sqrt{3}}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$18-12\sqrt{2}+\frac{3\sqrt{6}+2\sqrt{6}}{\sqrt{3}}$$

Combine 3\sqrt{6} and 2\sqrt{6} to get 5\sqrt{6}.

$$18-12\sqrt{2}+\frac{5\sqrt{6}}{\sqrt{3}}$$

Rationalize the denominator of \frac{5\sqrt{6}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$18-12\sqrt{2}+\frac{5\sqrt{6}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$18-12\sqrt{2}+\frac{5\sqrt{6}\sqrt{3}}{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$18-12\sqrt{2}+\frac{5\sqrt{3}\sqrt{2}\sqrt{3}}{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$18-12\sqrt{2}+\frac{5\times 3\sqrt{2}}{3}$$

Cancel out 3 and 3.

$$18-12\sqrt{2}+5\sqrt{2}$$

Combine -12\sqrt{2} and 5\sqrt{2} to get -7\sqrt{2}.

$$18-7\sqrt{2}$$