Solve (13/5+2sqrt{3})

$$( \frac { 13 } { 5 + 2 \sqrt { 3 } } )$$

Evaluate

$$5-2\sqrt{3}\approx 1.535898385$$

Solution Steps

Rationalize the denominator of \frac{13}{5+2\sqrt{3}} by multiplying numerator and denominator by 5-2\sqrt{3}.

$$\frac{13\left(5-2\sqrt{3}\right)}{\left(5+2\sqrt{3}\right)\left(5-2\sqrt{3}\right)}$$

Consider \left(5+2\sqrt{3}\right)\left(5-2\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{13\left(5-2\sqrt{3}\right)}{5^{2}-\left(2\sqrt{3}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{13\left(5-2\sqrt{3}\right)}{25-\left(2\sqrt{3}\right)^{2}}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\frac{13\left(5-2\sqrt{3}\right)}{25-2^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{13\left(5-2\sqrt{3}\right)}{25-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{13\left(5-2\sqrt{3}\right)}{25-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{13\left(5-2\sqrt{3}\right)}{25-12}$$

Subtract 12 from 25 to get 13.

$$\frac{13\left(5-2\sqrt{3}\right)}{13}$$

Cancel out 13 and 13.

$$5-2\sqrt{3}$$