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

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

Evaluate

$$\sqrt{2}-1\approx 0.414213562$$

Solution Steps

Divide \sqrt{2}-1 by \sqrt{2}-1 to get 1.

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

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

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

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}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

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

$$\sqrt{2}-1$$