Solve frac{sqrt{50}-sqrt{3}}{2sqrt{3}}-frac{10-sqrt{6}}{2sqrt{6}}

$$\frac { \sqrt { 50 } - \sqrt { 3 } } { 2 \sqrt { 3 } } - \frac { 10 - \sqrt { 6 } } { 2 \sqrt { 6 } }$$

Evaluate

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Short Solution Steps

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\frac{5\sqrt{2}-\sqrt{3}}{2\sqrt{3}}-\frac{10-\sqrt{6}}{2\sqrt{6}}$$

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

$$\frac{\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}-\frac{10-\sqrt{6}}{2\sqrt{6}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{2\times 3}-\frac{10-\sqrt{6}}{2\sqrt{6}}$$

Multiply 2 and 3 to get 6.

$$\frac{\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{6}-\frac{10-\sqrt{6}}{2\sqrt{6}}$$

Rationalize the denominator of \frac{10-\sqrt{6}}{2\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{6}-\frac{\left(10-\sqrt{6}\right)\sqrt{6}}{2\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{6}-\frac{\left(10-\sqrt{6}\right)\sqrt{6}}{2\times 6}$$

Multiply 2 and 6 to get 12.

$$\frac{\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{6}-\frac{\left(10-\sqrt{6}\right)\sqrt{6}}{12}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 12 is 12. Multiply \frac{\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{6} times \frac{2}{2}.

$$\frac{2\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{12}-\frac{\left(10-\sqrt{6}\right)\sqrt{6}}{12}$$

Since \frac{2\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}}{12} and \frac{\left(10-\sqrt{6}\right)\sqrt{6}}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}-\left(10-\sqrt{6}\right)\sqrt{6}}{12}$$

Do the multiplications in 2\left(5\sqrt{2}-\sqrt{3}\right)\sqrt{3}-\left(10-\sqrt{6}\right)\sqrt{6}.

$$\frac{10\sqrt{6}-6-10\sqrt{6}+6}{12}$$

Do the calculations in 10\sqrt{6}-6-10\sqrt{6}+6.

$$\frac{0}{12}$$

Zero divided by any non-zero number gives zero.

$$0$$

Factor

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