Solve sqrt{6/7}+7+7/6

$$\sqrt{ \frac{ 6 }{ 7 } } +7+ \frac{ 7 }{ 6 }$$

Evaluate

$$\frac{\sqrt{42}}{7}+\frac{49}{6}\approx 9.092486766$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{6}{7}} as the division of square roots \frac{\sqrt{6}}{\sqrt{7}}.

$$\frac{\sqrt{6}}{\sqrt{7}}+7+\frac{7}{6}$$

Rationalize the denominator of \frac{\sqrt{6}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\sqrt{6}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}+7+\frac{7}{6}$$

The square of \sqrt{7} is 7.

$$\frac{\sqrt{6}\sqrt{7}}{7}+7+\frac{7}{6}$$

To multiply \sqrt{6} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{42}}{7}+7+\frac{7}{6}$$

Convert 7 to fraction \frac{42}{6}.

$$\frac{\sqrt{42}}{7}+\frac{42}{6}+\frac{7}{6}$$

Since \frac{42}{6} and \frac{7}{6} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{42}}{7}+\frac{42+7}{6}$$

Add 42 and 7 to get 49.

$$\frac{\sqrt{42}}{7}+\frac{49}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 6 is 42. Multiply \frac{\sqrt{42}}{7} times \frac{6}{6}. Multiply \frac{49}{6} times \frac{7}{7}.

$$\frac{6\sqrt{42}}{42}+\frac{49\times 7}{42}$$

Since \frac{6\sqrt{42}}{42} and \frac{49\times 7}{42} have the same denominator, add them by adding their numerators.

$$\frac{6\sqrt{42}+49\times 7}{42}$$

Do the multiplications in 6\sqrt{42}+49\times 7.

$$\frac{6\sqrt{42}+343}{42}$$

Factor

$$\frac{6 \sqrt{42} + 343}{42} = 9.092486766439217$$