Solve 1/7-2sqrt{2+1}

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

Evaluate

$$\frac{\sqrt{2}+4}{28}\approx 0.19336477$$

Solution Steps

Add 7 and 1 to get 8.

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

Rationalize the denominator of \frac{1}{8-2\sqrt{2}} by multiplying numerator and denominator by 8+2\sqrt{2}.

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

Consider \left(8-2\sqrt{2}\right)\left(8+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}.

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

Calculate 8 to the power of 2 and get 64.

$$\frac{8+2\sqrt{2}}{64-\left(-2\sqrt{2}\right)^{2}}$$

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

$$\frac{8+2\sqrt{2}}{64-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{8+2\sqrt{2}}{64-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{8+2\sqrt{2}}{64-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{8+2\sqrt{2}}{64-8}$$

Subtract 8 from 64 to get 56.

$$\frac{8+2\sqrt{2}}{56}$$