Solve frac{sqrt{2}+1-1}{sqrt{2}+1}

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

Evaluate

$$2-\sqrt{2}\approx 0.585786438$$

Solution Steps

Subtract 1 from 1 to get 0.

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

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{2}+1} by multiplying numerator and denominator by \sqrt{2}-1.

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

Consider \left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

Use the distributive property to multiply \sqrt{2} by \sqrt{2}-1.

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

The square of \sqrt{2} is 2.

$$2-\sqrt{2}$$