Solve (frac{sqrt{3}}{sqrt{5}}-frac{sqrt{5}}{sqrt{2}})^2

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

Evaluate

$$\frac{31}{10}-\sqrt{6}\approx 0.650510257$$

Short Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 2 is 10. Multiply \frac{\sqrt{15}}{5} times \frac{2}{2}. Multiply \frac{\sqrt{10}}{2} times \frac{5}{5}.

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

Since \frac{2\sqrt{15}}{10} and \frac{5\sqrt{10}}{10} have the same denominator, subtract them by subtracting their numerators.

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

To raise \frac{2\sqrt{15}-5\sqrt{10}}{10} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2\sqrt{15}-5\sqrt{10}\right)^{2}.

$$\frac{4\left(\sqrt{15}\right)^{2}-20\sqrt{15}\sqrt{10}+25\left(\sqrt{10}\right)^{2}}{10^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{4\times 15-20\sqrt{15}\sqrt{10}+25\left(\sqrt{10}\right)^{2}}{10^{2}}$$

Multiply 4 and 15 to get 60.

$$\frac{60-20\sqrt{15}\sqrt{10}+25\left(\sqrt{10}\right)^{2}}{10^{2}}$$

To multiply \sqrt{15} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{60-20\sqrt{150}+25\left(\sqrt{10}\right)^{2}}{10^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{60-20\sqrt{150}+25\times 10}{10^{2}}$$

Multiply 25 and 10 to get 250.

$$\frac{60-20\sqrt{150}+250}{10^{2}}$$

Add 60 and 250 to get 310.

$$\frac{310-20\sqrt{150}}{10^{2}}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{310-20\times 5\sqrt{6}}{10^{2}}$$

Multiply -20 and 5 to get -100.

$$\frac{310-100\sqrt{6}}{10^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{310-100\sqrt{6}}{100}$$

Expand

$$\frac{31}{10} - \sqrt{6} = 0.650510257$$

Short Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 2 is 10. Multiply \frac{\sqrt{15}}{5} times \frac{2}{2}. Multiply \frac{\sqrt{10}}{2} times \frac{5}{5}.

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

Since \frac{2\sqrt{15}}{10} and \frac{5\sqrt{10}}{10} have the same denominator, subtract them by subtracting their numerators.

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

To raise \frac{2\sqrt{15}-5\sqrt{10}}{10} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2\sqrt{15}-5\sqrt{10}\right)^{2}.

$$\frac{4\left(\sqrt{15}\right)^{2}-20\sqrt{15}\sqrt{10}+25\left(\sqrt{10}\right)^{2}}{10^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{4\times 15-20\sqrt{15}\sqrt{10}+25\left(\sqrt{10}\right)^{2}}{10^{2}}$$

Multiply 4 and 15 to get 60.

$$\frac{60-20\sqrt{15}\sqrt{10}+25\left(\sqrt{10}\right)^{2}}{10^{2}}$$

To multiply \sqrt{15} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{60-20\sqrt{150}+25\left(\sqrt{10}\right)^{2}}{10^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{60-20\sqrt{150}+25\times 10}{10^{2}}$$

Multiply 25 and 10 to get 250.

$$\frac{60-20\sqrt{150}+250}{10^{2}}$$

Add 60 and 250 to get 310.

$$\frac{310-20\sqrt{150}}{10^{2}}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{310-20\times 5\sqrt{6}}{10^{2}}$$

Multiply -20 and 5 to get -100.

$$\frac{310-100\sqrt{6}}{10^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{310-100\sqrt{6}}{100}$$