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

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

Evaluate

$$\frac{2\sqrt{2}-1}{7}\approx 0.261203875$$

Solution Steps

Combine \sqrt{2} and 1\sqrt{2} to get 2\sqrt{2}.

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

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

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

Consider \left(2\sqrt{2}+1\right)\left(2\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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}-1}{4\times 2-1^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{2\sqrt{2}-1}{8-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

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

Subtract 1 from 8 to get 7.

$$\frac{2\sqrt{2}-1}{7}$$