Solve 5/-5-3sqrt{3}

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

Evaluate

$$\frac{25-15\sqrt{3}}{2}\approx -0.490381057$$

Solution Steps

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

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

Consider \left(-5-3\sqrt{3}\right)\left(-5+3\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}.

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

Calculate -5 to the power of 2 and get 25.

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

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

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

Calculate -3 to the power of 2 and get 9.

$$\frac{5\left(-5+3\sqrt{3}\right)}{25-9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5\left(-5+3\sqrt{3}\right)}{25-9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{5\left(-5+3\sqrt{3}\right)}{25-27}$$

Subtract 27 from 25 to get -2.

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

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

$$\frac{-25+15\sqrt{3}}{-2}$$