Solve frac{2+sqrt{2}}{2+sqrt{2}-2}

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

Evaluate

$$\sqrt{2}+1\approx 2.414213562$$

Solution Steps

Subtract 2 from 2 to get 0.

$$\frac{2+\sqrt{2}}{\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

$$\frac{\left(2+\sqrt{2}\right)\sqrt{2}}{2}$$

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

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

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}+2}{2}$$

Divide each term of 2\sqrt{2}+2 by 2 to get \sqrt{2}+1.

$$\sqrt{2}+1$$

Factor

$$\sqrt{2} + 1 = 2.414213562$$