Solve -3/5+2sqrt{3}

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

Evaluate

$$\frac{6\sqrt{3}-15}{13}\approx -0.354438089$$

Solution Steps

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

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

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}.

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

Calculate 5 to the power of 2 and get 25.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

Subtract 12 from 25 to get 13.

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

Use the distributive property to multiply 3 by 5-2\sqrt{3}.

$$-\frac{15-6\sqrt{3}}{13}$$