Solve 12/5div3/5-3/7div6-1/7+frac{sqrt{3frac{1}{5}}}{14}

$$\frac { 12 } { 5 } \div \frac { 3 } { 5 } - \frac { 3 } { 7 } \div 6 - \frac { 1 } { 7 } + \frac { \sqrt { 3 \frac { 1 } { 5 } } } { 14 }$$

Evaluate

$$\frac{2\sqrt{5}}{35}+\frac{53}{14}\approx 3.913489599$$

Short Solution Steps

Divide \frac{12}{5} by \frac{3}{5} by multiplying \frac{12}{5} by the reciprocal of \frac{3}{5}.

$$\frac{12}{5}\times \frac{5}{3}-\frac{\frac{3}{7}}{6}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Multiply \frac{12}{5} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{12\times 5}{5\times 3}-\frac{\frac{3}{7}}{6}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Cancel out 5 in both numerator and denominator.

$$\frac{12}{3}-\frac{\frac{3}{7}}{6}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Divide 12 by 3 to get 4.

$$4-\frac{\frac{3}{7}}{6}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Express \frac{\frac{3}{7}}{6} as a single fraction.

$$4-\frac{3}{7\times 6}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Multiply 7 and 6 to get 42.

$$4-\frac{3}{42}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Reduce the fraction \frac{3}{42} to lowest terms by extracting and canceling out 3.

$$4-\frac{1}{14}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Convert 4 to fraction \frac{56}{14}.

$$\frac{56}{14}-\frac{1}{14}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Since \frac{56}{14} and \frac{1}{14} have the same denominator, subtract them by subtracting their numerators.

$$\frac{56-1}{14}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Subtract 1 from 56 to get 55.

$$\frac{55}{14}-\frac{1}{7}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Least common multiple of 14 and 7 is 14. Convert \frac{55}{14} and \frac{1}{7} to fractions with denominator 14.

$$\frac{55}{14}-\frac{2}{14}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Since \frac{55}{14} and \frac{2}{14} have the same denominator, subtract them by subtracting their numerators.

$$\frac{55-2}{14}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Subtract 2 from 55 to get 53.

$$\frac{53}{14}+\frac{\sqrt{\frac{3\times 5+1}{5}}}{14}$$

Multiply 3 and 5 to get 15.

$$\frac{53}{14}+\frac{\sqrt{\frac{15+1}{5}}}{14}$$

Add 15 and 1 to get 16.

$$\frac{53}{14}+\frac{\sqrt{\frac{16}{5}}}{14}$$

Rewrite the square root of the division \sqrt{\frac{16}{5}} as the division of square roots \frac{\sqrt{16}}{\sqrt{5}}.

$$\frac{53}{14}+\frac{\frac{\sqrt{16}}{\sqrt{5}}}{14}$$

Calculate the square root of 16 and get 4.

$$\frac{53}{14}+\frac{\frac{4}{\sqrt{5}}}{14}$$

Rationalize the denominator of \frac{4}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{53}{14}+\frac{\frac{4\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}{14}$$

The square of \sqrt{5} is 5.

$$\frac{53}{14}+\frac{\frac{4\sqrt{5}}{5}}{14}$$

Express \frac{\frac{4\sqrt{5}}{5}}{14} as a single fraction.

$$\frac{53}{14}+\frac{4\sqrt{5}}{5\times 14}$$

Cancel out 2 in both numerator and denominator.

$$\frac{53}{14}+\frac{2\sqrt{5}}{5\times 7}$$

Multiply 5 and 7 to get 35.

$$\frac{53}{14}+\frac{2\sqrt{5}}{35}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 14 and 35 is 70. Multiply \frac{53}{14} times \frac{5}{5}. Multiply \frac{2\sqrt{5}}{35} times \frac{2}{2}.

$$\frac{53\times 5}{70}+\frac{2\times 2\sqrt{5}}{70}$$

Since \frac{53\times 5}{70} and \frac{2\times 2\sqrt{5}}{70} have the same denominator, add them by adding their numerators.

$$\frac{53\times 5+2\times 2\sqrt{5}}{70}$$

Do the multiplications in 53\times 5+2\times 2\sqrt{5}.

$$\frac{265+4\sqrt{5}}{70}$$

Factor

$$\frac{4 \sqrt{5} + 265}{70} = 3.9134895987142735$$