Solve (10/sqrt{5}-5/sqrt{3})(2/sqrt{3}+4/sqrt{5})

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

Evaluate

$$\frac{14}{3}\approx 4.666666667$$

Short Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Divide 10\sqrt{5} by 5 to get 2\sqrt{5}.

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

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

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

The square of \sqrt{3} is 3.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{5} times \frac{3}{3}.

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

Since \frac{3\times 2\sqrt{5}}{3} and \frac{5\sqrt{3}}{3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\times 2\sqrt{5}-5\sqrt{3}.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2}{\sqrt{3}}+\frac{4}{\sqrt{5}}\right)$$

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

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\frac{4}{\sqrt{5}}\right)$$

The square of \sqrt{3} is 3.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2\sqrt{3}}{3}+\frac{4}{\sqrt{5}}\right)$$

Rationalize the denominator of \frac{4}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2\sqrt{3}}{3}+\frac{4\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\right)$$

The square of \sqrt{5} is 5.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2\sqrt{3}}{3}+\frac{4\sqrt{5}}{5}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 5 is 15. Multiply \frac{2\sqrt{3}}{3} times \frac{5}{5}. Multiply \frac{4\sqrt{5}}{5} times \frac{3}{3}.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{5\times 2\sqrt{3}}{15}+\frac{3\times 4\sqrt{5}}{15}\right)$$

Since \frac{5\times 2\sqrt{3}}{15} and \frac{3\times 4\sqrt{5}}{15} have the same denominator, add them by adding their numerators.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\times \frac{5\times 2\sqrt{3}+3\times 4\sqrt{5}}{15}$$

Do the multiplications in 5\times 2\sqrt{3}+3\times 4\sqrt{5}.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\times \frac{10\sqrt{3}+12\sqrt{5}}{15}$$

Multiply \frac{6\sqrt{5}-5\sqrt{3}}{3} times \frac{10\sqrt{3}+12\sqrt{5}}{15} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(6\sqrt{5}-5\sqrt{3}\right)\left(10\sqrt{3}+12\sqrt{5}\right)}{3\times 15}$$

Multiply 3 and 15 to get 45.

$$\frac{\left(6\sqrt{5}-5\sqrt{3}\right)\left(10\sqrt{3}+12\sqrt{5}\right)}{45}$$

Apply the distributive property by multiplying each term of 6\sqrt{5}-5\sqrt{3} by each term of 10\sqrt{3}+12\sqrt{5}.

$$\frac{60\sqrt{3}\sqrt{5}+72\left(\sqrt{5}\right)^{2}-50\left(\sqrt{3}\right)^{2}-60\sqrt{3}\sqrt{5}}{45}$$

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

$$\frac{60\sqrt{15}+72\left(\sqrt{5}\right)^{2}-50\left(\sqrt{3}\right)^{2}-60\sqrt{3}\sqrt{5}}{45}$$

The square of \sqrt{5} is 5.

$$\frac{60\sqrt{15}+72\times 5-50\left(\sqrt{3}\right)^{2}-60\sqrt{3}\sqrt{5}}{45}$$

Multiply 72 and 5 to get 360.

$$\frac{60\sqrt{15}+360-50\left(\sqrt{3}\right)^{2}-60\sqrt{3}\sqrt{5}}{45}$$

The square of \sqrt{3} is 3.

$$\frac{60\sqrt{15}+360-50\times 3-60\sqrt{3}\sqrt{5}}{45}$$

Multiply -50 and 3 to get -150.

$$\frac{60\sqrt{15}[phone]\sqrt{3}\sqrt{5}}{45}$$

Subtract 150 from 360 to get 210.

$$\frac{60\sqrt{15}+210-60\sqrt{3}\sqrt{5}}{45}$$

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

$$\frac{60\sqrt{15}+210-60\sqrt{15}}{45}$$

Combine 60\sqrt{15} and -60\sqrt{15} to get 0.

$$\frac{210}{45}$$

Reduce the fraction \frac{210}{45} to lowest terms by extracting and canceling out 15.

$$\frac{14}{3}$$

Factor

$$\frac{2 \cdot 7}{3} = 4\frac{2}{3} = 4.666666666666667$$