Solve 1/sqrt{50}+6/sqrt{50}varepsilon

$$\frac { 1 } { \sqrt { 50 } } + \frac { 6 } { \sqrt { 50 } } \varepsilon$$

Evaluate

$$\frac{\sqrt{2}\left(6\epsilon +1\right)}{10}$$

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}.

$$\frac{1}{5\sqrt{2}}+\frac{6}{\sqrt{50}}\epsilon$$

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

$$\frac{\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}+\frac{6}{\sqrt{50}}\epsilon$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{5\times 2}+\frac{6}{\sqrt{50}}\epsilon$$

Multiply 5 and 2 to get 10.

$$\frac{\sqrt{2}}{10}+\frac{6}{\sqrt{50}}\epsilon$$

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}.

$$\frac{\sqrt{2}}{10}+\frac{6}{5\sqrt{2}}\epsilon$$

Rationalize the denominator of \frac{6}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{2}}{10}+\frac{6\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}\epsilon$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{10}+\frac{6\sqrt{2}}{5\times 2}\epsilon$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{2}}{10}+\frac{3\sqrt{2}}{5}\epsilon$$

Express \frac{3\sqrt{2}}{5}\epsilon as a single fraction.

$$\frac{\sqrt{2}}{10}+\frac{3\sqrt{2}\epsilon }{5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 10 and 5 is 10. Multiply \frac{3\sqrt{2}\epsilon }{5} times \frac{2}{2}.

$$\frac{\sqrt{2}}{10}+\frac{2\times 3\sqrt{2}\epsilon }{10}$$

Since \frac{\sqrt{2}}{10} and \frac{2\times 3\sqrt{2}\epsilon }{10} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{2}+2\times 3\sqrt{2}\epsilon }{10}$$

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

$$\frac{\sqrt{2}+6\sqrt{2}\epsilon }{10}$$

Factor

$$\frac{\sqrt{2}\left(6\epsilon +1\right)}{10}$$

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}.

$$factor(\frac{1}{5\sqrt{2}}+\frac{6}{\sqrt{50}}\epsilon )$$

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

$$factor(\frac{\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}+\frac{6}{\sqrt{50}}\epsilon )$$

The square of \sqrt{2} is 2.

$$factor(\frac{\sqrt{2}}{5\times 2}+\frac{6}{\sqrt{50}}\epsilon )$$

Multiply 5 and 2 to get 10.

$$factor(\frac{\sqrt{2}}{10}+\frac{6}{\sqrt{50}}\epsilon )$$

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}.

$$factor(\frac{\sqrt{2}}{10}+\frac{6}{5\sqrt{2}}\epsilon )$$

Rationalize the denominator of \frac{6}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$factor(\frac{\sqrt{2}}{10}+\frac{6\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}\epsilon )$$

The square of \sqrt{2} is 2.

$$factor(\frac{\sqrt{2}}{10}+\frac{6\sqrt{2}}{5\times 2}\epsilon )$$

Cancel out 2 in both numerator and denominator.

$$factor(\frac{\sqrt{2}}{10}+\frac{3\sqrt{2}}{5}\epsilon )$$

Express \frac{3\sqrt{2}}{5}\epsilon as a single fraction.

$$factor(\frac{\sqrt{2}}{10}+\frac{3\sqrt{2}\epsilon }{5})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 10 and 5 is 10. Multiply \frac{3\sqrt{2}\epsilon }{5} times \frac{2}{2}.

$$factor(\frac{\sqrt{2}}{10}+\frac{2\times 3\sqrt{2}\epsilon }{10})$$

Since \frac{\sqrt{2}}{10} and \frac{2\times 3\sqrt{2}\epsilon }{10} have the same denominator, add them by adding their numerators.

$$factor(\frac{\sqrt{2}+2\times 3\sqrt{2}\epsilon }{10})$$

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

$$factor(\frac{\sqrt{2}+6\sqrt{2}\epsilon }{10})$$

Consider \sqrt{2}+6\sqrt{2}\epsilon . Factor out \sqrt{2}.

$$\sqrt{2}\left(1+6\epsilon \right)$$

Rewrite the complete factored expression.

$$\frac{\left(6\epsilon +1\right)\sqrt{2}}{10}$$