Solve 2/5-sqrt{3}

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

Evaluate

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

Solution Steps

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

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

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

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

Square 5. Square \sqrt{3}.

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

Subtract 3 from 25 to get 22.

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

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

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

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

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

Multiply \frac{1}{11} and 5 to get \frac{5}{11}.

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