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

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

Evaluate

$$\frac{\sqrt{2}+1}{2}\approx 1.207106781$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2}-1\right)^{2}.

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

The square of \sqrt{2} is 2.

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

Add 2 and 1 to get 3.

$$\frac{\sqrt{2}}{3-2\sqrt{2}+1}$$

Add 3 and 1 to get 4.

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

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

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

Consider \left(4-2\sqrt{2}\right)\left(4+2\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{\sqrt{2}\left(4+2\sqrt{2}\right)}{16-\left(-2\sqrt{2}\right)^{2}}$$

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

$$\frac{\sqrt{2}\left(4+2\sqrt{2}\right)}{16-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\sqrt{2}\left(4+2\sqrt{2}\right)}{16-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}\left(4+2\sqrt{2}\right)}{16-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\sqrt{2}\left(4+2\sqrt{2}\right)}{16-8}$$

Subtract 8 from 16 to get 8.

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

Use the distributive property to multiply \sqrt{2} by 4+2\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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