Solve frac{sqrt{2}-sqrt{4}+3sqrt{2}+2}{(-1)^2+sqrt{8}-1-sqrt{2}}

$$\frac { \sqrt { 2 } - \sqrt { 4 } + 3 \sqrt { 2 } + 2 } { ( - 1 ) ^ { 2 } + \sqrt { 8 } - 1 - \sqrt { 2 } }$$

Evaluate

$$4$$

Solution Steps

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{2}-2+3\sqrt{2}+2}{\left(-1\right)^{2}+\sqrt{8}-1-\sqrt{2}}$$

Combine \sqrt{2} and 3\sqrt{2} to get 4\sqrt{2}.

$$\frac{4\sqrt{2}-2+2}{\left(-1\right)^{2}+\sqrt{8}-1-\sqrt{2}}$$

Add -2 and 2 to get 0.

$$\frac{4\sqrt{2}}{\left(-1\right)^{2}+\sqrt{8}-1-\sqrt{2}}$$

Calculate -1 to the power of 2 and get 1.

$$\frac{4\sqrt{2}}{1+\sqrt{8}-1-\sqrt{2}}$$

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}.

$$\frac{4\sqrt{2}}{1+2\sqrt{2}-1-\sqrt{2}}$$

Subtract 1 from 1 to get 0.

$$\frac{4\sqrt{2}}{2\sqrt{2}-\sqrt{2}}$$

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

$$\frac{4\sqrt{2}}{\sqrt{2}}$$

Rationalize the denominator of \frac{4\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{2}\sqrt{2}}{2}$$

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

$$\frac{4\times 2}{2}$$

Multiply 4 and 2 to get 8.

$$\frac{8}{2}$$

Divide 8 by 2 to get 4.

$$4$$

Factor

$$2^{2}$$