Solve 4/4+4{4+4/4+4{4+frac{4/4+frac{4{4+frac{4/4}}}}}}

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

Evaluate

$$\frac{169}{204}\approx 0.828431373$$

Solution Steps

Divide 4 by 4 to get 1.

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

Add 4 and 1 to get 5.

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

Convert 4 to fraction \frac{20}{5}.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{\frac{20}{5}+\frac{4}{5}}}}}}$$

Since \frac{20}{5} and \frac{4}{5} have the same denominator, add them by adding their numerators.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{\frac{20+4}{5}}}}}}$$

Add 20 and 4 to get 24.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{\frac{24}{5}}}}}}$$

Divide 4 by \frac{24}{5} by multiplying 4 by the reciprocal of \frac{24}{5}.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{4+4\times \frac{5}{24}}}}}$$

Express 4\times \frac{5}{24} as a single fraction.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4\times 5}{24}}}}}$$

Multiply 4 and 5 to get 20.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{20}{24}}}}}$$

Reduce the fraction \frac{20}{24} to lowest terms by extracting and canceling out 4.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{5}{6}}}}}$$

Convert 4 to fraction \frac{24}{6}.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{\frac{24}{6}+\frac{5}{6}}}}}$$

Since \frac{24}{6} and \frac{5}{6} have the same denominator, add them by adding their numerators.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{\frac{24+5}{6}}}}}$$

Add 24 and 5 to get 29.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4}{\frac{29}{6}}}}}$$

Divide 4 by \frac{29}{6} by multiplying 4 by the reciprocal of \frac{29}{6}.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+4\times \frac{6}{29}}}}$$

Express 4\times \frac{6}{29} as a single fraction.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{4\times 6}{29}}}}$$

Multiply 4 and 6 to get 24.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{24}{29}}}}$$

Convert 4 to fraction \frac{116}{29}.

$$\frac{4}{4+\frac{4}{4+\frac{4}{\frac{116}{29}+\frac{24}{29}}}}$$

Since \frac{116}{29} and \frac{24}{29} have the same denominator, add them by adding their numerators.

$$\frac{4}{4+\frac{4}{4+\frac{4}{\frac{116+24}{29}}}}$$

Add 116 and 24 to get 140.

$$\frac{4}{4+\frac{4}{4+\frac{4}{\frac{140}{29}}}}$$

Divide 4 by \frac{140}{29} by multiplying 4 by the reciprocal of \frac{140}{29}.

$$\frac{4}{4+\frac{4}{4+4\times \frac{29}{140}}}$$

Express 4\times \frac{29}{140} as a single fraction.

$$\frac{4}{4+\frac{4}{4+\frac{4\times 29}{140}}}$$

Multiply 4 and 29 to get 116.

$$\frac{4}{4+\frac{4}{4+\frac{116}{140}}}$$

Reduce the fraction \frac{116}{140} to lowest terms by extracting and canceling out 4.

$$\frac{4}{4+\frac{4}{4+\frac{29}{35}}}$$

Convert 4 to fraction \frac{140}{35}.

$$\frac{4}{4+\frac{4}{\frac{140}{35}+\frac{29}{35}}}$$

Since \frac{140}{35} and \frac{29}{35} have the same denominator, add them by adding their numerators.

$$\frac{4}{4+\frac{4}{\frac{140+29}{35}}}$$

Add 140 and 29 to get 169.

$$\frac{4}{4+\frac{4}{\frac{169}{35}}}$$

Divide 4 by \frac{169}{35} by multiplying 4 by the reciprocal of \frac{169}{35}.

$$\frac{4}{4+4\times \frac{35}{169}}$$

Express 4\times \frac{35}{169} as a single fraction.

$$\frac{4}{4+\frac{4\times 35}{169}}$$

Multiply 4 and 35 to get 140.

$$\frac{4}{4+\frac{140}{169}}$$

Convert 4 to fraction \frac{676}{169}.

$$\frac{4}{\frac{676}{169}+\frac{140}{169}}$$

Since \frac{676}{169} and \frac{140}{169} have the same denominator, add them by adding their numerators.

$$\frac{4}{\frac{676+140}{169}}$$

Add 676 and 140 to get 816.

$$\frac{4}{\frac{816}{169}}$$

Divide 4 by \frac{816}{169} by multiplying 4 by the reciprocal of \frac{816}{169}.

$$4\times \frac{169}{816}$$

Express 4\times \frac{169}{816} as a single fraction.

$$\frac{4\times 169}{816}$$

Multiply 4 and 169 to get 676.

$$\frac{676}{816}$$

Reduce the fraction \frac{676}{816} to lowest terms by extracting and canceling out 4.

$$\frac{169}{204}$$

Factor

$$\frac{13 ^ {2}}{2 ^ {2} \cdot 3 \cdot 17} = 0.8284313725490197$$