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}}}}}}}}}}

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

Evaluate

$$\frac{4756}{5741}\approx 0.828427103$$

Solution Steps

Divide 4 by 4 to get 1.

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

Add 4 and 1 to get 5.

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

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

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

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

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

Add 20 and 4 to get 24.

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

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

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

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

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

Multiply 4 and 5 to get 20.

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

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

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

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

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

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

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

Add 24 and 5 to get 29.

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

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

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

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

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

Multiply 4 and 6 to get 24.

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

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

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

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

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

Add 116 and 24 to get 140.

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

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

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

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

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

Multiply 4 and 29 to get 116.

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

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

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

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

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

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

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

Add 140 and 29 to get 169.

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

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

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

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

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

Multiply 4 and 35 to get 140.

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

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

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

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

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

Add 676 and 140 to get 816.

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

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

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

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

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

Multiply 4 and 169 to get 676.

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{676}{816}}}}$$

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

$$\frac{4}{4+\frac{4}{4+\frac{4}{4+\frac{169}{204}}}}$$

Convert 4 to fraction \frac{816}{204}.

$$\frac{4}{4+\frac{4}{4+\frac{4}{\frac{816}{204}+\frac{169}{204}}}}$$

Since \frac{816}{204} and \frac{169}{204} have the same denominator, add them by adding their numerators.

$$\frac{4}{4+\frac{4}{4+\frac{4}{\frac{816+169}{204}}}}$$

Add 816 and 169 to get 985.

$$\frac{4}{4+\frac{4}{4+\frac{4}{\frac{985}{204}}}}$$

Divide 4 by \frac{985}{204} by multiplying 4 by the reciprocal of \frac{985}{204}.

$$\frac{4}{4+\frac{4}{4+4\times \frac{204}{985}}}$$

Express 4\times \frac{204}{985} as a single fraction.

$$\frac{4}{4+\frac{4}{4+\frac{4\times 204}{985}}}$$

Multiply 4 and 204 to get 816.

$$\frac{4}{4+\frac{4}{4+\frac{816}{985}}}$$

Convert 4 to fraction \frac{3940}{985}.

$$\frac{4}{4+\frac{4}{\frac{3940}{985}+\frac{816}{985}}}$$

Since \frac{3940}{985} and \frac{816}{985} have the same denominator, add them by adding their numerators.

$$\frac{4}{4+\frac{4}{\frac{3940+816}{985}}}$$

Add 3940 and 816 to get 4756.

$$\frac{4}{4+\frac{4}{\frac{4756}{985}}}$$

Divide 4 by \frac{4756}{985} by multiplying 4 by the reciprocal of \frac{4756}{985}.

$$\frac{4}{4+4\times \frac{985}{4756}}$$

Express 4\times \frac{985}{4756} as a single fraction.

$$\frac{4}{4+\frac{4\times 985}{4756}}$$

Multiply 4 and 985 to get 3940.

$$\frac{4}{4+\frac{3940}{4756}}$$

Reduce the fraction \frac{3940}{4756} to lowest terms by extracting and canceling out 4.

$$\frac{4}{4+\frac{985}{1189}}$$

Convert 4 to fraction \frac{4756}{1189}.

$$\frac{4}{\frac{4756}{1189}+\frac{985}{1189}}$$

Since \frac{4756}{1189} and \frac{985}{1189} have the same denominator, add them by adding their numerators.

$$\frac{4}{\frac{4756+985}{1189}}$$

Add 4756 and 985 to get 5741.

$$\frac{4}{\frac{5741}{1189}}$$

Divide 4 by \frac{5741}{1189} by multiplying 4 by the reciprocal of \frac{5741}{1189}.

$$4\times \frac{1189}{5741}$$

Express 4\times \frac{1189}{5741} as a single fraction.

$$\frac{4\times 1189}{5741}$$

Multiply 4 and 1189 to get 4756.

$$\frac{4756}{5741}$$

Factor

$$\frac{2 ^ {2} \cdot 29 \cdot 41}{5741} = 0.8284271032921093$$