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

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

Evaluate

$$\frac{\sqrt{2}+4}{7}\approx 0.77345908$$

Solution Steps

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

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

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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

$$\frac{\sqrt{2}\left(2\sqrt{2}+1\right)}{8-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\sqrt{2}\left(2\sqrt{2}+1\right)}{8-1}$$

Subtract 1 from 8 to get 7.

$$\frac{\sqrt{2}\left(2\sqrt{2}+1\right)}{7}$$

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

$$\frac{2\left(\sqrt{2}\right)^{2}+\sqrt{2}}{7}$$

The square of \sqrt{2} is 2.

$$\frac{2\times 2+\sqrt{2}}{7}$$

Multiply 2 and 2 to get 4.

$$\frac{4+\sqrt{2}}{7}$$