Solve 14/5sqrt{3-sqrt{5}}

$$\frac{ 14 }{ 5 \sqrt{ 3 } - \sqrt{ 5 } }$$

Evaluate

$$\frac{\sqrt{5}\left(\sqrt{15}+1\right)}{5}\approx 2.179264403$$

Solution Steps

Rationalize the denominator of \frac{14}{5\sqrt{3}-\sqrt{5}} by multiplying numerator and denominator by 5\sqrt{3}+\sqrt{5}.

$$\frac{14\left(5\sqrt{3}+\sqrt{5}\right)}{\left(5\sqrt{3}-\sqrt{5}\right)\left(5\sqrt{3}+\sqrt{5}\right)}$$

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

$$\frac{14\left(5\sqrt{3}+\sqrt{5}\right)}{\left(5\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

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

$$\frac{14\left(5\sqrt{3}+\sqrt{5}\right)}{5^{2}\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{14\left(5\sqrt{3}+\sqrt{5}\right)}{25\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{14\left(5\sqrt{3}+\sqrt{5}\right)}{25\times 3-\left(\sqrt{5}\right)^{2}}$$

Multiply 25 and 3 to get 75.

$$\frac{14\left(5\sqrt{3}+\sqrt{5}\right)}{75-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{14\left(5\sqrt{3}+\sqrt{5}\right)}{75-5}$$

Subtract 5 from 75 to get 70.

$$\frac{14\left(5\sqrt{3}+\sqrt{5}\right)}{70}$$

Divide 14\left(5\sqrt{3}+\sqrt{5}\right) by 70 to get \frac{1}{5}\left(5\sqrt{3}+\sqrt{5}\right).

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

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

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

Cancel out 5 and 5.

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