Solve 2/sqrt{3-5}

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

Evaluate

$$\frac{-\sqrt{3}-5}{11}\approx -0.612004619$$

Solution Steps

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

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

Consider \left(\sqrt{3}-5\right)\left(\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}.

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

Square \sqrt{3}. Square 5.

$$\frac{2\left(\sqrt{3}+5\right)}{3-25}$$

Subtract 25 from 3 to get -22.

$$\frac{2\left(\sqrt{3}+5\right)}{-22}$$

Divide 2\left(\sqrt{3}+5\right) by -22 to get -\frac{1}{11}\left(\sqrt{3}+5\right).

$$-\frac{1}{11}\left(\sqrt{3}+5\right)$$

Use the distributive property to multiply -\frac{1}{11} by \sqrt{3}+5.

$$-\frac{1}{11}\sqrt{3}-\frac{1}{11}\times 5$$

Express -\frac{1}{11}\times 5 as a single fraction.

$$-\frac{1}{11}\sqrt{3}+\frac{-5}{11}$$

Fraction \frac{-5}{11} can be rewritten as -\frac{5}{11} by extracting the negative sign.

$$-\frac{1}{11}\sqrt{3}-\frac{5}{11}$$