Solve 1/5sqrt{3-3sqrt{5}}

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

Evaluate

$$\frac{3\sqrt{5}+5\sqrt{3}}{30}\approx 0.512281932$$

Solution Steps

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

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

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

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

Expand \left(5\sqrt{3}\right)^{2}.

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

Calculate 5 to the power of 2 and get 25.

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

The square of \sqrt{3} is 3.

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

Multiply 25 and 3 to get 75.

$$\frac{5\sqrt{3}+3\sqrt{5}}{75-\left(-3\sqrt{5}\right)^{2}}$$

Expand \left(-3\sqrt{5}\right)^{2}.

$$\frac{5\sqrt{3}+3\sqrt{5}}{75-\left(-3\right)^{2}\left(\sqrt{5}\right)^{2}}$$

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

$$\frac{5\sqrt{3}+3\sqrt{5}}{75-9\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{5\sqrt{3}+3\sqrt{5}}{75-9\times 5}$$

Multiply 9 and 5 to get 45.

$$\frac{5\sqrt{3}+3\sqrt{5}}{75-45}$$

Subtract 45 from 75 to get 30.

$$\frac{5\sqrt{3}+3\sqrt{5}}{30}$$

Factor

$$\frac{\sqrt{3} \sqrt{5} {(\sqrt{3} + \sqrt{5})}}{30} = 0.5122819323447919$$