Solve 1/sqrt{2}-1/sqrt{3}-1/sqrt{2-5}

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

Evaluate

$$\frac{25\sqrt{2}}{46}-\frac{\sqrt{3}}{3}+\frac{5}{23}\approx 0.408635363$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

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

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

The square of \sqrt{3} is 3.

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

Rationalize the denominator of \frac{1}{\sqrt{2}-5} by multiplying numerator and denominator by \sqrt{2}+5.

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

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

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

Square \sqrt{2}. Square 5.

$$\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{3}-\frac{\sqrt{2}+5}{2-25}$$

Subtract 25 from 2 to get -23.

$$\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{3}-\frac{\sqrt{2}+5}{-23}$$

Multiply both numerator and denominator by -1.

$$\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{3}-\frac{-\sqrt{2}-5}{23}$$

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

$$\frac{3\sqrt{2}}{6}-\frac{2\sqrt{3}}{6}-\frac{-\sqrt{2}-5}{23}$$

Since \frac{3\sqrt{2}}{6} and \frac{2\sqrt{3}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\sqrt{2}-2\sqrt{3}}{6}-\frac{-\sqrt{2}-5}{23}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 23 is 138. Multiply \frac{3\sqrt{2}-2\sqrt{3}}{6} times \frac{23}{23}. Multiply \frac{-\sqrt{2}-5}{23} times \frac{6}{6}.

$$\frac{23\left(3\sqrt{2}-2\sqrt{3}\right)}{138}-\frac{6\left(-\sqrt{2}-5\right)}{138}$$

Since \frac{23\left(3\sqrt{2}-2\sqrt{3}\right)}{138} and \frac{6\left(-\sqrt{2}-5\right)}{138} have the same denominator, subtract them by subtracting their numerators.

$$\frac{23\left(3\sqrt{2}-2\sqrt{3}\right)-6\left(-\sqrt{2}-5\right)}{138}$$

Do the multiplications in 23\left(3\sqrt{2}-2\sqrt{3}\right)-6\left(-\sqrt{2}-5\right).

$$\frac{69\sqrt{2}-46\sqrt{3}+6\sqrt{2}+30}{138}$$

Do the calculations in 69\sqrt{2}-46\sqrt{3}+6\sqrt{2}+30.

$$\frac{75\sqrt{2}-46\sqrt{3}+30}{138}$$