Solve 2/2+2{2+frac{2/2+frac{2{2+2}}}}

$$\frac { 2 } { 2 + \frac { 2 } { 2 + \frac { 2 } { 2 + \frac { 2 } { 2 + 2 } } } }$$

Evaluate

$$\frac{14}{19}\approx 0.736842105$$

Solution Steps

Add 2 and 2 to get 4.

$$\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{4}}}}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{1}{2}}}}$$

Convert 2 to fraction \frac{4}{2}.

$$\frac{2}{2+\frac{2}{2+\frac{2}{\frac{4}{2}+\frac{1}{2}}}}$$

Since \frac{4}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{2}{2+\frac{2}{2+\frac{2}{\frac{4+1}{2}}}}$$

Add 4 and 1 to get 5.

$$\frac{2}{2+\frac{2}{2+\frac{2}{\frac{5}{2}}}}$$

Divide 2 by \frac{5}{2} by multiplying 2 by the reciprocal of \frac{5}{2}.

$$\frac{2}{2+\frac{2}{2+2\times \frac{2}{5}}}$$

Express 2\times \frac{2}{5} as a single fraction.

$$\frac{2}{2+\frac{2}{2+\frac{2\times 2}{5}}}$$

Multiply 2 and 2 to get 4.

$$\frac{2}{2+\frac{2}{2+\frac{4}{5}}}$$

Convert 2 to fraction \frac{10}{5}.

$$\frac{2}{2+\frac{2}{\frac{10}{5}+\frac{4}{5}}}$$

Since \frac{10}{5} and \frac{4}{5} have the same denominator, add them by adding their numerators.

$$\frac{2}{2+\frac{2}{\frac{10+4}{5}}}$$

Add 10 and 4 to get 14.

$$\frac{2}{2+\frac{2}{\frac{14}{5}}}$$

Divide 2 by \frac{14}{5} by multiplying 2 by the reciprocal of \frac{14}{5}.

$$\frac{2}{2+2\times \frac{5}{14}}$$

Express 2\times \frac{5}{14} as a single fraction.

$$\frac{2}{2+\frac{2\times 5}{14}}$$

Multiply 2 and 5 to get 10.

$$\frac{2}{2+\frac{10}{14}}$$

Reduce the fraction \frac{10}{14} to lowest terms by extracting and canceling out 2.

$$\frac{2}{2+\frac{5}{7}}$$

Convert 2 to fraction \frac{14}{7}.

$$\frac{2}{\frac{14}{7}+\frac{5}{7}}$$

Since \frac{14}{7} and \frac{5}{7} have the same denominator, add them by adding their numerators.

$$\frac{2}{\frac{14+5}{7}}$$

Add 14 and 5 to get 19.

$$\frac{2}{\frac{19}{7}}$$

Divide 2 by \frac{19}{7} by multiplying 2 by the reciprocal of \frac{19}{7}.

$$2\times \frac{7}{19}$$

Express 2\times \frac{7}{19} as a single fraction.

$$\frac{2\times 7}{19}$$

Multiply 2 and 7 to get 14.

$$\frac{14}{19}$$

Factor

$$\frac{2 \cdot 7}{19} = 0.7368421052631579$$