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

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

Evaluate

$$\frac{3\sqrt{15}+7}{17}\approx 1.095232355$$

Short Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{20-\left(\sqrt{3}\right)^{2}}-\frac{\sqrt{5}}{\sqrt{3}-2\sqrt{5}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{20-3}-\frac{\sqrt{5}}{\sqrt{3}-2\sqrt{5}}$$

Subtract 3 from 20 to get 17.

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}}{\sqrt{3}-2\sqrt{5}}$$

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

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{\left(\sqrt{3}-2\sqrt{5}\right)\left(\sqrt{3}+2\sqrt{5}\right)}$$

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

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{\left(\sqrt{3}\right)^{2}-\left(-2\sqrt{5}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{3-\left(-2\sqrt{5}\right)^{2}}$$

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

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{3-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{3-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{3-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{3-20}$$

Subtract 20 from 3 to get -17.

$$\frac{\sqrt{3}\left(2\sqrt{5}-\sqrt{3}\right)}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{-17}$$

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

$$\frac{2\sqrt{3}\sqrt{5}-\left(\sqrt{3}\right)^{2}}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{-17}$$

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

$$\frac{2\sqrt{15}-\left(\sqrt{3}\right)^{2}}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{-17}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{15}-3}{17}-\frac{\sqrt{5}\left(\sqrt{3}+2\sqrt{5}\right)}{-17}$$

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

$$\frac{2\sqrt{15}-3}{17}-\frac{\sqrt{5}\sqrt{3}+2\left(\sqrt{5}\right)^{2}}{-17}$$

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

$$\frac{2\sqrt{15}-3}{17}-\frac{\sqrt{15}+2\left(\sqrt{5}\right)^{2}}{-17}$$

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{15}-3}{17}-\frac{\sqrt{15}+2\times 5}{-17}$$

Multiply 2 and 5 to get 10.

$$\frac{2\sqrt{15}-3}{17}-\frac{\sqrt{15}+10}{-17}$$

Multiply both numerator and denominator by -1.

$$\frac{2\sqrt{15}-3}{17}-\frac{-\sqrt{15}-10}{17}$$

Since \frac{2\sqrt{15}-3}{17} and \frac{-\sqrt{15}-10}{17} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\sqrt{15}-3-\left(-\sqrt{15}-10\right)}{17}$$

Do the multiplications in 2\sqrt{15}-3-\left(-\sqrt{15}-10\right).

$$\frac{2\sqrt{15}-3+\sqrt{15}+10}{17}$$

Do the calculations in 2\sqrt{15}-3+\sqrt{15}+10.

$$\frac{3\sqrt{15}+7}{17}$$