Solve 5/7+sqrt{54/5}

$$\frac { 5 } { 7 } + \sqrt { 5 \frac { 4 } { 5 } }$$

Evaluate

$$\frac{\sqrt{145}}{5}+\frac{5}{7}\approx 3.12260463$$

Short Solution Steps

Multiply 5 and 5 to get 25.

$$\frac{5}{7}+\sqrt{\frac{25+4}{5}}$$

Add 25 and 4 to get 29.

$$\frac{5}{7}+\sqrt{\frac{29}{5}}$$

Rewrite the square root of the division \sqrt{\frac{29}{5}} as the division of square roots \frac{\sqrt{29}}{\sqrt{5}}.

$$\frac{5}{7}+\frac{\sqrt{29}}{\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{29}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{5}{7}+\frac{\sqrt{29}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{5}{7}+\frac{\sqrt{29}\sqrt{5}}{5}$$

To multiply \sqrt{29} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{5}{7}+\frac{\sqrt{145}}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 5 is 35. Multiply \frac{5}{7} times \frac{5}{5}. Multiply \frac{\sqrt{145}}{5} times \frac{7}{7}.

$$\frac{5\times 5}{35}+\frac{7\sqrt{145}}{35}$$

Since \frac{5\times 5}{35} and \frac{7\sqrt{145}}{35} have the same denominator, add them by adding their numerators.

$$\frac{5\times 5+7\sqrt{145}}{35}$$

Do the multiplications in 5\times 5+7\sqrt{145}.

$$\frac{25+7\sqrt{145}}{35}$$

Factor

$$\frac{7 \sqrt{145} + 25}{35} = 3.122604630044173$$