Solve frac{5*sqrt{3}}{2-2*sqrt{3}}

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

Evaluate

$$\frac{-5\sqrt{3}-15}{4}\approx -5.915063509$$

Solution Steps

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

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

Consider \left(2-2\sqrt{3}\right)\left(2+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}.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

Calculate -2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

$$\frac{5\sqrt{3}\left(2+2\sqrt{3}\right)}{4-12}$$

Subtract 12 from 4 to get -8.

$$\frac{5\sqrt{3}\left(2+2\sqrt{3}\right)}{-8}$$

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

$$\frac{10\sqrt{3}+10\left(\sqrt{3}\right)^{2}}{-8}$$

The square of \sqrt{3} is 3.

$$\frac{10\sqrt{3}+10\times 3}{-8}$$

Multiply 10 and 3 to get 30.

$$\frac{10\sqrt{3}+30}{-8}$$