Solve frac{2000sqrt{3}}{3-sqrt{3}}

$$\frac{ 2000 \sqrt{ 3 } }{ 3- \sqrt{ 3 } }$$

Evaluate

$$1000\left(\sqrt{3}+1\right)\approx 2732.050807569$$

Solution Steps

Rationalize the denominator of \frac{2000\sqrt{3}}{3-\sqrt{3}} by multiplying numerator and denominator by 3+\sqrt{3}.

$$\frac{2000\sqrt{3}\left(3+\sqrt{3}\right)}{\left(3-\sqrt{3}\right)\left(3+\sqrt{3}\right)}$$

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

$$\frac{2000\sqrt{3}\left(3+\sqrt{3}\right)}{3^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 3. Square \sqrt{3}.

$$\frac{2000\sqrt{3}\left(3+\sqrt{3}\right)}{9-3}$$

Subtract 3 from 9 to get 6.

$$\frac{2000\sqrt{3}\left(3+\sqrt{3}\right)}{6}$$

Use the distributive property to multiply 2000\sqrt{3} by 3+\sqrt{3}.

$$\frac{6000\sqrt{3}+2000\left(\sqrt{3}\right)^{2}}{6}$$

The square of \sqrt{3} is 3.

$$\frac{6000\sqrt{3}+2000\times 3}{6}$$

Multiply 2000 and 3 to get 6000.

$$\frac{6000\sqrt{3}+6000}{6}$$