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

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

Evaluate

$$4$$

Solution Steps

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

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

Calculate -\sqrt{2} to the power of 2 and get \left(\sqrt{2}\right)^{2}.

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

Multiply -2 and -1 to get 2.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\sqrt{\left(\sqrt{2}\right)^{2}+2\times 2+\left(\sqrt{2}\right)^{2}+\left(\sqrt{2}-\left(-\sqrt{2}\right)\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{\left(\sqrt{2}\right)^{2}+2\times 2+2+\left(\sqrt{2}-\left(-\sqrt{2}\right)\right)^{2}}$$

Combine 2\times 2 and 2 to get 3\times 2.

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

Multiply 3 and 2 to get 6.

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

The square of \sqrt{2} is 2.

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

Add 2 and 6 to get 8.

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

Multiply -1 and -1 to get 1.

$$\sqrt{8+\left(\sqrt{2}+\sqrt{2}\right)^{2}}$$

Combine \sqrt{2} and \sqrt{2} to get 2\sqrt{2}.

$$\sqrt{8+\left(2\sqrt{2}\right)^{2}}$$

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

$$\sqrt{8+2^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{8+4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{8+4\times 2}$$

Multiply 4 and 2 to get 8.

$$\sqrt{8+8}$$

Add 8 and 8 to get 16.

$$\sqrt{16}$$

Calculate the square root of 16 and get 4.

$$4$$

Factor

$$2^{2}$$