Solve 14/2sqrt{3-sqrt{5}}

$$\frac { 14 } { 2 \sqrt { 3 } - \sqrt { 5 } }$$

Evaluate

$$2\left(\sqrt{5}+2\sqrt{3}\right)\approx 11.400339185$$

Solution Steps

Rationalize the denominator of \frac{14}{2\sqrt{3}-\sqrt{5}} by multiplying numerator and denominator by 2\sqrt{3}+\sqrt{5}.

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{14\left(2\sqrt{3}+\sqrt{5}\right)}{4\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{14\left(2\sqrt{3}+\sqrt{5}\right)}{4\times 3-\left(\sqrt{5}\right)^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{14\left(2\sqrt{3}+\sqrt{5}\right)}{12-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{14\left(2\sqrt{3}+\sqrt{5}\right)}{12-5}$$

Subtract 5 from 12 to get 7.

$$\frac{14\left(2\sqrt{3}+\sqrt{5}\right)}{7}$$

Divide 14\left(2\sqrt{3}+\sqrt{5}\right) by 7 to get 2\left(2\sqrt{3}+\sqrt{5}\right).

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

Use the distributive property to multiply 2 by 2\sqrt{3}+\sqrt{5}.

$$4\sqrt{3}+2\sqrt{5}$$