Solve 1/1-2sqrt{3}

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

Evaluate

$$\frac{-2\sqrt{3}-1}{11}\approx -0.40582742$$

Solution Steps

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

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

Consider \left(1-2\sqrt{3}\right)\left(1+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}.

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

Calculate 1 to the power of 2 and get 1.

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

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

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

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

$$\frac{1+2\sqrt{3}}{1-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{1+2\sqrt{3}}{1-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{1+2\sqrt{3}}{1-12}$$

Subtract 12 from 1 to get -11.

$$\frac{1+2\sqrt{3}}{-11}$$

Multiply both numerator and denominator by -1.

$$\frac{-1-2\sqrt{3}}{11}$$