Solve (sqrt{6}-3)^2-(sqrt{6}-2)(sqrt{16}+2)+2sqrt{54}

$$( \sqrt { 6 } - 3 ) ^ { 2 } - ( \sqrt { 6 } - 2 ) ( \sqrt { 16 } + 2 ) + 2 \sqrt { 54 }$$

Evaluate

$$27-6\sqrt{6}\approx 12.303061543$$

Solution Steps

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

$$\left(\sqrt{6}\right)^{2}-6\sqrt{6}+9-\left(\sqrt{6}-2\right)\left(\sqrt{16}+2\right)+2\sqrt{54}$$

The square of \sqrt{6} is 6.

$$6-6\sqrt{6}+9-\left(\sqrt{6}-2\right)\left(\sqrt{16}+2\right)+2\sqrt{54}$$

Add 6 and 9 to get 15.

$$15-6\sqrt{6}-\left(\sqrt{6}-2\right)\left(\sqrt{16}+2\right)+2\sqrt{54}$$

Calculate the square root of 16 and get 4.

$$15-6\sqrt{6}-\left(\sqrt{6}-2\right)\left(4+2\right)+2\sqrt{54}$$

Add 4 and 2 to get 6.

$$15-6\sqrt{6}-\left(\sqrt{6}-2\right)\times 6+2\sqrt{54}$$

Use the distributive property to multiply \sqrt{6}-2 by 6.

$$15-6\sqrt{6}-\left(6\sqrt{6}-12\right)+2\sqrt{54}$$

To find the opposite of 6\sqrt{6}-12, find the opposite of each term.

$$15-6\sqrt{6}-6\sqrt{6}+12+2\sqrt{54}$$

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

$$15-12\sqrt{6}+12+2\sqrt{54}$$

Add 15 and 12 to get 27.

$$27-12\sqrt{6}+2\sqrt{54}$$

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}.

$$27-12\sqrt{6}+2\times 3\sqrt{6}$$

Multiply 2 and 3 to get 6.

$$27-12\sqrt{6}+6\sqrt{6}$$

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

$$27-6\sqrt{6}$$