Solve frac{sqrt{5}+sqrt{3}}{2sqrt{3}-sqrt{5}}

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

Evaluate

$$\frac{3\sqrt{15}+11}{7}\approx 3.231278577$$

Solution Steps

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

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

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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

The square of \sqrt{5} is 5.

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

Subtract 5 from 12 to get 7.

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

Apply the distributive property by multiplying each term of \sqrt{5}+\sqrt{3} by each term of 2\sqrt{3}+\sqrt{5}.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{2\sqrt{15}+\left(\sqrt{5}\right)^{2}+2\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}}{7}$$

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{15}+5+2\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}}{7}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{15}+5+2\times 3+\sqrt{3}\sqrt{5}}{7}$$

Multiply 2 and 3 to get 6.

$$\frac{2\sqrt{15}+5+6+\sqrt{3}\sqrt{5}}{7}$$

Add 5 and 6 to get 11.

$$\frac{2\sqrt{15}+11+\sqrt{3}\sqrt{5}}{7}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{2\sqrt{15}+11+\sqrt{15}}{7}$$

Combine 2\sqrt{15} and \sqrt{15} to get 3\sqrt{15}.

$$\frac{3\sqrt{15}+11}{7}$$