Solve frac{2sqrt{3}}{1-3sqrt{3}}

$$\frac { 2 \sqrt { 3 } } { 1 - 3 \sqrt { 3 } }$$

Evaluate

$$\frac{-\sqrt{3}-9}{13}\approx -0.82554237$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{3}}{1-3\sqrt{3}} by multiplying numerator and denominator by 1+3\sqrt{3}.

$$\frac{2\sqrt{3}\left(1+3\sqrt{3}\right)}{\left(1-3\sqrt{3}\right)\left(1+3\sqrt{3}\right)}$$

Consider \left(1-3\sqrt{3}\right)\left(1+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}.

$$\frac{2\sqrt{3}\left(1+3\sqrt{3}\right)}{1^{2}-\left(-3\sqrt{3}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{2\sqrt{3}\left(1+3\sqrt{3}\right)}{1-\left(-3\sqrt{3}\right)^{2}}$$

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

$$\frac{2\sqrt{3}\left(1+3\sqrt{3}\right)}{1-\left(-3\right)^{2}\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{2\sqrt{3}\left(1+3\sqrt{3}\right)}{1-9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}\left(1+3\sqrt{3}\right)}{1-9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{2\sqrt{3}\left(1+3\sqrt{3}\right)}{1-27}$$

Subtract 27 from 1 to get -26.

$$\frac{2\sqrt{3}\left(1+3\sqrt{3}\right)}{-26}$$

Divide 2\sqrt{3}\left(1+3\sqrt{3}\right) by -26 to get -\frac{1}{13}\sqrt{3}\left(1+3\sqrt{3}\right).

$$-\frac{1}{13}\sqrt{3}\left(1+3\sqrt{3}\right)$$

Use the distributive property to multiply -\frac{1}{13}\sqrt{3} by 1+3\sqrt{3}.

$$-\frac{1}{13}\sqrt{3}-\frac{1}{13}\sqrt{3}\times 3\sqrt{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$-\frac{1}{13}\sqrt{3}-\frac{1}{13}\times 3\times 3$$

Express -\frac{1}{13}\times 3 as a single fraction.

$$-\frac{1}{13}\sqrt{3}+\frac{-3}{13}\times 3$$

Fraction \frac{-3}{13} can be rewritten as -\frac{3}{13} by extracting the negative sign.

$$-\frac{1}{13}\sqrt{3}-\frac{3}{13}\times 3$$

Express -\frac{3}{13}\times 3 as a single fraction.

$$-\frac{1}{13}\sqrt{3}+\frac{-3\times 3}{13}$$

Multiply -3 and 3 to get -9.

$$-\frac{1}{13}\sqrt{3}+\frac{-9}{13}$$

Fraction \frac{-9}{13} can be rewritten as -\frac{9}{13} by extracting the negative sign.

$$-\frac{1}{13}\sqrt{3}-\frac{9}{13}$$