Solve 1/2sqrt{3-5}

$$\frac { 1 } { 2 \sqrt { 3 } - 5 }$$

Evaluate

$$\frac{-2\sqrt{3}-5}{13}\approx -0.65108474$$

Solution Steps

Rationalize the denominator of \frac{1}{2\sqrt{3}-5} by multiplying numerator and denominator by 2\sqrt{3}+5.

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

Consider \left(2\sqrt{3}-5\right)\left(2\sqrt{3}+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}+5}{4\times 3-5^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{2\sqrt{3}+5}{12-5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{2\sqrt{3}+5}{12-25}$$

Subtract 25 from 12 to get -13.

$$\frac{2\sqrt{3}+5}{-13}$$

Multiply both numerator and denominator by -1.

$$\frac{-2\sqrt{3}-5}{13}$$