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

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

Evaluate

$$-\frac{\sqrt{15}}{7}+\frac{5}{14}\approx -0.196140478$$

Short Solution Steps

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

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

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

$$\frac{\frac{\sqrt{5}-\sqrt{3}}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}}{\frac{1}{\sqrt{5}}-\sqrt{3}}$$

Square \sqrt{5}. Square \sqrt{3}.

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

Subtract 3 from 5 to get 2.

$$\frac{\frac{\sqrt{5}-\sqrt{3}}{2}}{\frac{1}{\sqrt{5}}-\sqrt{3}}$$

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

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

The square of \sqrt{5} is 5.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{3} times \frac{5}{5}.

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

Since \frac{\sqrt{5}}{5} and \frac{5\sqrt{3}}{5} have the same denominator, subtract them by subtracting their numerators.

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

Divide \frac{\sqrt{5}-\sqrt{3}}{2} by \frac{\sqrt{5}-5\sqrt{3}}{5} by multiplying \frac{\sqrt{5}-\sqrt{3}}{2} by the reciprocal of \frac{\sqrt{5}-5\sqrt{3}}{5}.

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

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

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

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

$$\frac{5\sqrt{5}-5\sqrt{3}}{2\sqrt{5}-10\sqrt{3}}$$

Rationalize the denominator of \frac{5\sqrt{5}-5\sqrt{3}}{2\sqrt{5}-10\sqrt{3}} by multiplying numerator and denominator by 2\sqrt{5}+10\sqrt{3}.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{\left(2\sqrt{5}-10\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}$$

Consider \left(2\sqrt{5}-10\sqrt{3}\right)\left(2\sqrt{5}+10\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}.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{\left(2\sqrt{5}\right)^{2}-\left(-10\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{2^{2}\left(\sqrt{5}\right)^{2}-\left(-10\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{4\left(\sqrt{5}\right)^{2}-\left(-10\sqrt{3}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{4\times 5-\left(-10\sqrt{3}\right)^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-\left(-10\sqrt{3}\right)^{2}}$$

Expand \left(-10\sqrt{3}\right)^{2}.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-\left(-10\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -10 to the power of 2 and get 100.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-100\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-100\times 3}$$

Multiply 100 and 3 to get 300.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-300}$$

Subtract 300 from 20 to get -280.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{-280}$$

Apply the distributive property by multiplying each term of 5\sqrt{5}-5\sqrt{3} by each term of 2\sqrt{5}+10\sqrt{3}.

$$\frac{10\left(\sqrt{5}\right)^{2}+50\sqrt{3}\sqrt{5}-10\sqrt{3}\sqrt{5}-50\left(\sqrt{3}\right)^{2}}{-280}$$

The square of \sqrt{5} is 5.

$$\frac{10\times 5+50\sqrt{3}\sqrt{5}-10\sqrt{3}\sqrt{5}-50\left(\sqrt{3}\right)^{2}}{-280}$$

Multiply 10 and 5 to get 50.

$$\frac{50+50\sqrt{3}\sqrt{5}-10\sqrt{3}\sqrt{5}-50\left(\sqrt{3}\right)^{2}}{-280}$$

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

$$\frac{50+50\sqrt{15}-10\sqrt{3}\sqrt{5}-50\left(\sqrt{3}\right)^{2}}{-280}$$

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

$$\frac{50+50\sqrt{15}-10\sqrt{15}-50\left(\sqrt{3}\right)^{2}}{-280}$$

Combine 50\sqrt{15} and -10\sqrt{15} to get 40\sqrt{15}.

$$\frac{50+40\sqrt{15}-50\left(\sqrt{3}\right)^{2}}{-280}$$

The square of \sqrt{3} is 3.

$$\frac{50+40\sqrt{15}-50\times 3}{-280}$$

Multiply -50 and 3 to get -150.

$$\frac{50+40\sqrt{15}-150}{-280}$$

Subtract 150 from 50 to get -100.

$$\frac{-100+40\sqrt{15}}{-280}$$