Solve (sqrt{2}-12)^2-(6-sqrt{8})^2

$$( \sqrt { 2 } - 12 ) ^ { 2 } - ( 6 - \sqrt { 8 } ) ^ { 2 }$$

Evaluate

$$102$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2}-12\right)^{2}.

$$\left(\sqrt{2}\right)^{2}-24\sqrt{2}+144-\left(6-\sqrt{8}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2-24\sqrt{2}+144-\left(6-\sqrt{8}\right)^{2}$$

Add 2 and 144 to get 146.

$$146-24\sqrt{2}-\left(6-\sqrt{8}\right)^{2}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$146-24\sqrt{2}-\left(6-2\sqrt{2}\right)^{2}$$

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

$$146-24\sqrt{2}-\left(36-24\sqrt{2}+4\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$146-24\sqrt{2}-\left(36-24\sqrt{2}+4\times 2\right)$$

Multiply 4 and 2 to get 8.

$$146-24\sqrt{2}-\left(36-24\sqrt{2}+8\right)$$

Add 36 and 8 to get 44.

$$146-24\sqrt{2}-\left(44-24\sqrt{2}\right)$$

To find the opposite of 44-24\sqrt{2}, find the opposite of each term.

$$146-24\sqrt{2}-44+24\sqrt{2}$$

Subtract 44 from 146 to get 102.

$$102-24\sqrt{2}+24\sqrt{2}$$

Combine -24\sqrt{2} and 24\sqrt{2} to get 0.

$$102$$

Factor

$$2\times 3\times 17$$