Solve 2/2sqrt{6+1}

$$\frac { 2 } { 2 \sqrt { 6 } + 1 }$$

Evaluate

$$\frac{4\sqrt{6}-2}{23}\approx 0.339041694$$

Solution Steps

Rationalize the denominator of \frac{2}{2\sqrt{6}+1} by multiplying numerator and denominator by 2\sqrt{6}-1.

$$\frac{2\left(2\sqrt{6}-1\right)}{\left(2\sqrt{6}+1\right)\left(2\sqrt{6}-1\right)}$$

Consider \left(2\sqrt{6}+1\right)\left(2\sqrt{6}-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}.

$$\frac{2\left(2\sqrt{6}-1\right)}{\left(2\sqrt{6}\right)^{2}-1^{2}}$$

Expand \left(2\sqrt{6}\right)^{2}.

$$\frac{2\left(2\sqrt{6}-1\right)}{2^{2}\left(\sqrt{6}\right)^{2}-1^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\left(2\sqrt{6}-1\right)}{4\left(\sqrt{6}\right)^{2}-1^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{2\left(2\sqrt{6}-1\right)}{4\times 6-1^{2}}$$

Multiply 4 and 6 to get 24.

$$\frac{2\left(2\sqrt{6}-1\right)}{24-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{2\left(2\sqrt{6}-1\right)}{24-1}$$

Subtract 1 from 24 to get 23.

$$\frac{2\left(2\sqrt{6}-1\right)}{23}$$

Use the distributive property to multiply 2 by 2\sqrt{6}-1.

$$\frac{4\sqrt{6}-2}{23}$$