Solve 1/5+2sqrt{6}

$$\frac { 1 } { 5 + 2 \sqrt { 6 } }$$

Evaluate

$$5-2\sqrt{6}\approx 0.101020514$$

Solution Steps

Rationalize the denominator of \frac{1}{5+2\sqrt{6}} by multiplying numerator and denominator by 5-2\sqrt{6}.

$$\frac{5-2\sqrt{6}}{\left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right)}$$

Consider \left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{5-2\sqrt{6}}{5^{2}-\left(2\sqrt{6}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{5-2\sqrt{6}}{25-\left(2\sqrt{6}\right)^{2}}$$

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

$$\frac{5-2\sqrt{6}}{25-2^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5-2\sqrt{6}}{25-4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{5-2\sqrt{6}}{25-4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{5-2\sqrt{6}}{25-24}$$

Subtract 24 from 25 to get 1.

$$\frac{5-2\sqrt{6}}{1}$$

Anything divided by one gives itself.

$$5-2\sqrt{6}$$