Solve sqrt{1+1/4^2+1/5^2}+sqrt{1+1/5^2+1/6^2}+sqrt{1+frac{1}{{6}^{2}}+frac{1}{{7}^{2}}}+sqrt{1+frac{1}{{7}^{2}}+frac{1}{{8}^{2}}}+sqrt{1+frac{1}{{8}^{2}}+frac{1}{{9}^{2}}}+sqrt{1+frac{1}{{9}^2}+frac{1}{{10}^2}}+sqrt{1+frac{1}{{10}^2}+frac{1}{{11}^2}}

$$\sqrt{ 1+ \frac{ 1 }{ { 4 }^{ 2 } } + \frac{ 1 }{ { 5 }^{ 2 } } } + \sqrt{ 1+ \frac{ 1 }{ { 5 }^{ 2 } } + \frac{ 1 }{ { 6 }^{ 2 } } } + \sqrt{ 1+ \frac{ 1 }{ { 6 }^{ 2 } } + \frac{ 1 }{ { 7 }^{ 2 } } } + \sqrt{ 1+ \frac{ 1 }{ { 7 }^{ 2 } } + \frac{ 1 }{ { 8 }^{ 2 } } } + \sqrt{ 1+ \frac{ 1 }{ { 8 }^{ 2 } } + \frac{ 1 }{ { 9 }^{ 2 } } } + \sqrt{ 1+ \frac{ 1 }{ { 9 }^{ 2 } } + \frac{ 1 }{ { 10 }^{ 2 } } } + \sqrt{ 1+ \frac{ 1 }{ { 10 }^{ 2 } } + \frac{ 1 }{ { 11 }^{ 2 } } }$$

Evaluate

$$\frac{315}{44}\approx 7.159090909$$

Solution Steps

Calculate 4 to the power of 2 and get 16.

$$\sqrt{1+\frac{1}{16}+\frac{1}{5^{2}}}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Convert 1 to fraction \frac{16}{16}.

$$\sqrt{\frac{16}{16}+\frac{1}{16}+\frac{1}{5^{2}}}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{16}{16} and \frac{1}{16} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{16+1}{16}+\frac{1}{5^{2}}}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 16 and 1 to get 17.

$$\sqrt{\frac{17}{16}+\frac{1}{5^{2}}}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{\frac{17}{16}+\frac{1}{25}}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 16 and 25 is 400. Convert \frac{17}{16} and \frac{1}{25} to fractions with denominator 400.

$$\sqrt{\frac{425}{400}+\frac{16}{400}}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{425}{400} and \frac{16}{400} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{425+16}{400}}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 425 and 16 to get 441.

$$\sqrt{\frac{441}{400}}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Rewrite the square root of the division \frac{441}{400} as the division of square roots \frac{\sqrt{441}}{\sqrt{400}}. Take the square root of both numerator and denominator.

$$\frac{21}{20}+\sqrt{1+\frac{1}{5^{2}}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{21}{20}+\sqrt{1+\frac{1}{25}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Convert 1 to fraction \frac{25}{25}.

$$\frac{21}{20}+\sqrt{\frac{25}{25}+\frac{1}{25}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{25}{25} and \frac{1}{25} have the same denominator, add them by adding their numerators.

$$\frac{21}{20}+\sqrt{\frac{25+1}{25}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 25 and 1 to get 26.

$$\frac{21}{20}+\sqrt{\frac{26}{25}+\frac{1}{6^{2}}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{21}{20}+\sqrt{\frac{26}{25}+\frac{1}{36}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 25 and 36 is 900. Convert \frac{26}{25} and \frac{1}{36} to fractions with denominator 900.

$$\frac{21}{20}+\sqrt{\frac{936}{900}+\frac{25}{900}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{936}{900} and \frac{25}{900} have the same denominator, add them by adding their numerators.

$$\frac{21}{20}+\sqrt{\frac{936+25}{900}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 936 and 25 to get 961.

$$\frac{21}{20}+\sqrt{\frac{961}{900}}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Rewrite the square root of the division \frac{961}{900} as the division of square roots \frac{\sqrt{961}}{\sqrt{900}}. Take the square root of both numerator and denominator.

$$\frac{21}{20}+\frac{31}{30}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 20 and 30 is 60. Convert \frac{21}{20} and \frac{31}{30} to fractions with denominator 60.

$$\frac{63}{60}+\frac{62}{60}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{63}{60} and \frac{62}{60} have the same denominator, add them by adding their numerators.

$$\frac{63+62}{60}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 63 and 62 to get 125.

$$\frac{125}{60}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Reduce the fraction \frac{125}{60} to lowest terms by extracting and canceling out 5.

$$\frac{25}{12}+\sqrt{1+\frac{1}{6^{2}}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{25}{12}+\sqrt{1+\frac{1}{36}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Convert 1 to fraction \frac{36}{36}.

$$\frac{25}{12}+\sqrt{\frac{36}{36}+\frac{1}{36}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{36}{36} and \frac{1}{36} have the same denominator, add them by adding their numerators.

$$\frac{25}{12}+\sqrt{\frac{36+1}{36}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 36 and 1 to get 37.

$$\frac{25}{12}+\sqrt{\frac{37}{36}+\frac{1}{7^{2}}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{25}{12}+\sqrt{\frac{37}{36}+\frac{1}{49}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 36 and 49 is 1764. Convert \frac{37}{36} and \frac{1}{49} to fractions with denominator 1764.

$$\frac{25}{12}+\sqrt{\frac{1813}{1764}+\frac{36}{1764}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{1813}{1764} and \frac{36}{1764} have the same denominator, add them by adding their numerators.

$$\frac{25}{12}+\sqrt{\frac{1813+36}{1764}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 1813 and 36 to get 1849.

$$\frac{25}{12}+\sqrt{\frac{1849}{1764}}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Rewrite the square root of the division \frac{1849}{1764} as the division of square roots \frac{\sqrt{1849}}{\sqrt{1764}}. Take the square root of both numerator and denominator.

$$\frac{25}{12}+\frac{43}{42}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 12 and 42 is 84. Convert \frac{25}{12} and \frac{43}{42} to fractions with denominator 84.

$$\frac{175}{84}+\frac{86}{84}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{175}{84} and \frac{86}{84} have the same denominator, add them by adding their numerators.

$$\frac{175+86}{84}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 175 and 86 to get 261.

$$\frac{261}{84}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Reduce the fraction \frac{261}{84} to lowest terms by extracting and canceling out 3.

$$\frac{87}{28}+\sqrt{1+\frac{1}{7^{2}}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{87}{28}+\sqrt{1+\frac{1}{49}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Convert 1 to fraction \frac{49}{49}.

$$\frac{87}{28}+\sqrt{\frac{49}{49}+\frac{1}{49}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{49}{49} and \frac{1}{49} have the same denominator, add them by adding their numerators.

$$\frac{87}{28}+\sqrt{\frac{49+1}{49}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 49 and 1 to get 50.

$$\frac{87}{28}+\sqrt{\frac{50}{49}+\frac{1}{8^{2}}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{87}{28}+\sqrt{\frac{50}{49}+\frac{1}{64}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 49 and 64 is 3136. Convert \frac{50}{49} and \frac{1}{64} to fractions with denominator 3136.

$$\frac{87}{28}+\sqrt{\frac{3200}{3136}+\frac{49}{3136}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{3200}{3136} and \frac{49}{3136} have the same denominator, add them by adding their numerators.

$$\frac{87}{28}+\sqrt{\frac{3200+49}{3136}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 3200 and 49 to get 3249.

$$\frac{87}{28}+\sqrt{\frac{3249}{3136}}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Rewrite the square root of the division \frac{3249}{3136} as the division of square roots \frac{\sqrt{3249}}{\sqrt{3136}}. Take the square root of both numerator and denominator.

$$\frac{87}{28}+\frac{57}{56}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 28 and 56 is 56. Convert \frac{87}{28} and \frac{57}{56} to fractions with denominator 56.

$$\frac{174}{56}+\frac{57}{56}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{174}{56} and \frac{57}{56} have the same denominator, add them by adding their numerators.

$$\frac{174+57}{56}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 174 and 57 to get 231.

$$\frac{231}{56}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Reduce the fraction \frac{231}{56} to lowest terms by extracting and canceling out 7.

$$\frac{33}{8}+\sqrt{1+\frac{1}{8^{2}}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{33}{8}+\sqrt{1+\frac{1}{64}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Convert 1 to fraction \frac{64}{64}.

$$\frac{33}{8}+\sqrt{\frac{64}{64}+\frac{1}{64}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{64}{64} and \frac{1}{64} have the same denominator, add them by adding their numerators.

$$\frac{33}{8}+\sqrt{\frac{64+1}{64}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 64 and 1 to get 65.

$$\frac{33}{8}+\sqrt{\frac{65}{64}+\frac{1}{9^{2}}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{33}{8}+\sqrt{\frac{65}{64}+\frac{1}{81}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 64 and 81 is 5184. Convert \frac{65}{64} and \frac{1}{81} to fractions with denominator 5184.

$$\frac{33}{8}+\sqrt{\frac{5265}{5184}+\frac{64}{5184}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{5265}{5184} and \frac{64}{5184} have the same denominator, add them by adding their numerators.

$$\frac{33}{8}+\sqrt{\frac{5265+64}{5184}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 5265 and 64 to get 5329.

$$\frac{33}{8}+\sqrt{\frac{5329}{5184}}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Rewrite the square root of the division \frac{5329}{5184} as the division of square roots \frac{\sqrt{5329}}{\sqrt{5184}}. Take the square root of both numerator and denominator.

$$\frac{33}{8}+\frac{73}{72}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 8 and 72 is 72. Convert \frac{33}{8} and \frac{73}{72} to fractions with denominator 72.

$$\frac{297}{72}+\frac{73}{72}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{297}{72} and \frac{73}{72} have the same denominator, add them by adding their numerators.

$$\frac{297+73}{72}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 297 and 73 to get 370.

$$\frac{370}{72}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Reduce the fraction \frac{370}{72} to lowest terms by extracting and canceling out 2.

$$\frac{185}{36}+\sqrt{1+\frac{1}{9^{2}}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{185}{36}+\sqrt{1+\frac{1}{81}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Convert 1 to fraction \frac{81}{81}.

$$\frac{185}{36}+\sqrt{\frac{81}{81}+\frac{1}{81}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{81}{81} and \frac{1}{81} have the same denominator, add them by adding their numerators.

$$\frac{185}{36}+\sqrt{\frac{81+1}{81}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 81 and 1 to get 82.

$$\frac{185}{36}+\sqrt{\frac{82}{81}+\frac{1}{10^{2}}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{185}{36}+\sqrt{\frac{82}{81}+\frac{1}{100}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 81 and 100 is 8100. Convert \frac{82}{81} and \frac{1}{100} to fractions with denominator 8100.

$$\frac{185}{36}+\sqrt{\frac{8200}{8100}+\frac{81}{8100}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{8200}{8100} and \frac{81}{8100} have the same denominator, add them by adding their numerators.

$$\frac{185}{36}+\sqrt{\frac{8200+81}{8100}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 8200 and 81 to get 8281.

$$\frac{185}{36}+\sqrt{\frac{8281}{8100}}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Rewrite the square root of the division \frac{8281}{8100} as the division of square roots \frac{\sqrt{8281}}{\sqrt{8100}}. Take the square root of both numerator and denominator.

$$\frac{185}{36}+\frac{91}{90}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Least common multiple of 36 and 90 is 180. Convert \frac{185}{36} and \frac{91}{90} to fractions with denominator 180.

$$\frac{925}{180}+\frac{182}{180}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Since \frac{925}{180} and \frac{182}{180} have the same denominator, add them by adding their numerators.

$$\frac{925+182}{180}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Add 925 and 182 to get 1107.

$$\frac{1107}{180}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Reduce the fraction \frac{1107}{180} to lowest terms by extracting and canceling out 9.

$$\frac{123}{20}+\sqrt{1+\frac{1}{10^{2}}+\frac{1}{11^{2}}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{123}{20}+\sqrt{1+\frac{1}{100}+\frac{1}{11^{2}}}$$

Convert 1 to fraction \frac{100}{100}.

$$\frac{123}{20}+\sqrt{\frac{100}{100}+\frac{1}{100}+\frac{1}{11^{2}}}$$

Since \frac{100}{100} and \frac{1}{100} have the same denominator, add them by adding their numerators.

$$\frac{123}{20}+\sqrt{\frac{100+1}{100}+\frac{1}{11^{2}}}$$

Add 100 and 1 to get 101.

$$\frac{123}{20}+\sqrt{\frac{101}{100}+\frac{1}{11^{2}}}$$

Calculate 11 to the power of 2 and get 121.

$$\frac{123}{20}+\sqrt{\frac{101}{100}+\frac{1}{121}}$$

Least common multiple of 100 and 121 is 12100. Convert \frac{101}{100} and \frac{1}{121} to fractions with denominator 12100.

$$\frac{123}{20}+\sqrt{\frac{12221}{12100}+\frac{100}{12100}}$$

Since \frac{12221}{12100} and \frac{100}{12100} have the same denominator, add them by adding their numerators.

$$\frac{123}{20}+\sqrt{\frac{12221+100}{12100}}$$

Add 12221 and 100 to get 12321.

$$\frac{123}{20}+\sqrt{\frac{12321}{12100}}$$

Rewrite the square root of the division \frac{12321}{12100} as the division of square roots \frac{\sqrt{12321}}{\sqrt{12100}}. Take the square root of both numerator and denominator.

$$\frac{123}{20}+\frac{111}{110}$$

Least common multiple of 20 and 110 is 220. Convert \frac{123}{20} and \frac{111}{110} to fractions with denominator 220.

$$\frac{1353}{220}+\frac{222}{220}$$

Since \frac{1353}{220} and \frac{222}{220} have the same denominator, add them by adding their numerators.

$$\frac{1353+222}{220}$$

Add 1353 and 222 to get 1575.

$$\frac{1575}{220}$$

Reduce the fraction \frac{1575}{220} to lowest terms by extracting and canceling out 5.

$$\frac{315}{44}$$

Factor

$$\frac{3 ^ {2} \cdot 5 \cdot 7}{2 ^ {2} \cdot 11} = 7\frac{7}{44} = 7.159090909090909$$