Solve sqrt{8.2+18+330+13+330+750+22/frac{1{8.2}+1/18+1/330+1/13+frac{1}{330}+frac{1}{750}+frac{1}{22}}}

$$\sqrt{ \frac{ 8.2+18+330+13+330+750+22 }{ \frac{ 1 }{ 8.2 } + \frac{ 1 }{ 18 } + \frac{ 1 }{ 330 } + \frac{ 1 }{ 13 } + \frac{ 1 }{ 330 } + \frac{ 1 }{ 750 } + \frac{ 1 }{ 22 } } }$$

Evaluate

$$\frac{30\sqrt{87410977099446}}{4053539}\approx 69.194230882$$

Solution Steps

Add 8.2 and 18 to get 26.2.

$$\sqrt{\frac{26.2+330+13+330+750+22}{\frac{1}{8.2}+\frac{1}{18}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Add 26.2 and 330 to get 356.2.

$$\sqrt{\frac{356.2+13+330+750+22}{\frac{1}{8.2}+\frac{1}{18}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Add 356.2 and 13 to get 369.2.

$$\sqrt{\frac{369.2+330+750+22}{\frac{1}{8.2}+\frac{1}{18}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Add 369.2 and 330 to get 699.2.

$$\sqrt{\frac{699.2+750+22}{\frac{1}{8.2}+\frac{1}{18}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Add 699.2 and 750 to get 1449.2.

$$\sqrt{\frac{1449.2+22}{\frac{1}{8.2}+\frac{1}{18}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Add 1449.2 and 22 to get 1471.2.

$$\sqrt{\frac{1471.2}{\frac{1}{8.2}+\frac{1}{18}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Expand \frac{1}{8.2} by multiplying both numerator and the denominator by 10.

$$\sqrt{\frac{1471.2}{\frac{10}{82}+\frac{1}{18}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Reduce the fraction \frac{10}{82} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{1471.2}{\frac{5}{41}+\frac{1}{18}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Least common multiple of 41 and 18 is 738. Convert \frac{5}{41} and \frac{1}{18} to fractions with denominator 738.

$$\sqrt{\frac{1471.2}{\frac{90}{738}+\frac{41}{738}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Since \frac{90}{738} and \frac{41}{738} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1471.2}{\frac{90+41}{738}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Add 90 and 41 to get 131.

$$\sqrt{\frac{1471.2}{\frac{131}{738}+\frac{1}{330}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Least common multiple of 738 and 330 is 40590. Convert \frac{131}{738} and \frac{1}{330} to fractions with denominator 40590.

$$\sqrt{\frac{1471.2}{\frac{7205}{40590}+\frac{123}{40590}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Since \frac{7205}{40590} and \frac{123}{40590} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1471.2}{\frac{7205+123}{40590}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Add 7205 and 123 to get 7328.

$$\sqrt{\frac{1471.2}{\frac{7328}{40590}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Reduce the fraction \frac{7328}{40590} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{1471.2}{\frac{3664}{20295}+\frac{1}{13}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Least common multiple of 20295 and 13 is 263835. Convert \frac{3664}{20295} and \frac{1}{13} to fractions with denominator 263835.

$$\sqrt{\frac{1471.2}{\frac{47632}{263835}+\frac{20295}{263835}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Since \frac{47632}{263835} and \frac{20295}{263835} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1471.2}{\frac{47632+20295}{263835}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Add 47632 and 20295 to get 67927.

$$\sqrt{\frac{1471.2}{\frac{67927}{263835}+\frac{1}{330}+\frac{1}{750}+\frac{1}{22}}}$$

Least common multiple of 263835 and 330 is 527670. Convert \frac{67927}{263835} and \frac{1}{330} to fractions with denominator 527670.

$$\sqrt{\frac{1471.2}{\frac{135854}{527670}+\frac{1599}{527670}+\frac{1}{750}+\frac{1}{22}}}$$

Since \frac{135854}{527670} and \frac{1599}{527670} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1471.2}{\frac{135854+1599}{527670}+\frac{1}{750}+\frac{1}{22}}}$$

Add 135854 and 1599 to get 137453.

$$\sqrt{\frac{1471.2}{\frac{137453}{527670}+\frac{1}{750}+\frac{1}{22}}}$$

Least common multiple of 527670 and 750 is 13191750. Convert \frac{137453}{527670} and \frac{1}{750} to fractions with denominator 13191750.

$$\sqrt{\frac{1471.2}{\frac{3436325}{13191750}+\frac{17589}{13191750}+\frac{1}{22}}}$$

Since \frac{3436325}{13191750} and \frac{17589}{13191750} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1471.2}{\frac{3436325+17589}{13191750}+\frac{1}{22}}}$$

Add 3436325 and 17589 to get 3453914.

$$\sqrt{\frac{1471.2}{\frac{3453914}{13191750}+\frac{1}{22}}}$$

Reduce the fraction \frac{3453914}{13191750} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{1471.2}{\frac{1726957}{6595875}+\frac{1}{22}}}$$

Least common multiple of 6595875 and 22 is 13191750. Convert \frac{1726957}{6595875} and \frac{1}{22} to fractions with denominator 13191750.

$$\sqrt{\frac{1471.2}{\frac{3453914}{13191750}+\frac{599625}{13191750}}}$$

Since \frac{3453914}{13191750} and \frac{599625}{13191750} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1471.2}{\frac{3453914+599625}{13191750}}}$$

Add 3453914 and 599625 to get 4053539.

$$\sqrt{\frac{1471.2}{\frac{4053539}{13191750}}}$$

Divide 1471.2 by \frac{4053539}{13191750} by multiplying 1471.2 by the reciprocal of \frac{4053539}{13191750}.

$$\sqrt{1471.2\times \frac{13191750}{4053539}}$$

Convert decimal number 1471.2 to fraction \frac{14712}{10}. Reduce the fraction \frac{14712}{10} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{7356}{5}\times \frac{13191750}{4053539}}$$

Multiply \frac{7356}{5} times \frac{13191750}{4053539} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{7356\times 13191750}{5\times 4053539}}$$

Do the multiplications in the fraction \frac{7356\times 13191750}{5\times 4053539}.

$$\sqrt{\frac{97038513000}{20267695}}$$

Reduce the fraction \frac{97038513000}{20267695} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{19407702600}{4053539}}$$

Rewrite the square root of the division \sqrt{\frac{19407702600}{4053539}} as the division of square roots \frac{\sqrt{19407702600}}{\sqrt{4053539}}.

$$\frac{\sqrt{19407702600}}{\sqrt{4053539}}$$

Factor 19407702600=30^{2}\times 21564114. Rewrite the square root of the product \sqrt{30^{2}\times 21564114} as the product of square roots \sqrt{30^{2}}\sqrt{21564114}. Take the square root of 30^{2}.

$$\frac{30\sqrt{21564114}}{\sqrt{4053539}}$$

Rationalize the denominator of \frac{30\sqrt{21564114}}{\sqrt{4053539}} by multiplying numerator and denominator by \sqrt{4053539}.

$$\frac{30\sqrt{21564114}\sqrt{4053539}}{\left(\sqrt{4053539}\right)^{2}}$$

The square of \sqrt{4053539} is 4053539.

$$\frac{30\sqrt{21564114}\sqrt{4053539}}{4053539}$$

To multiply \sqrt{21564114} and \sqrt{4053539}, multiply the numbers under the square root.

$$\frac{30\sqrt{87410977099446}}{4053539}$$