Solve (1-1/11^{2})*(1-1/12^{2})*(1-1/13^2)*(1-1/14^2)*(1-frac{1}{15^2})*(1-frac{1}{16^2})

$$( 1 - \frac { 1 } { 11 ^ { 2 } } ) \cdot ( 1 - \frac { 1 } { 12 ^ { 2 } } ) \cdot ( 1 - \frac { 1 } { 13 ^ { 2 } } ) \cdot ( 1 - \frac { 1 } { 14 ^ { 2 } } ) \cdot ( 1 - \frac { 1 } { 15 ^ { 2 } } ) \cdot ( 1 - \frac { 1 } { 16 ^ { 2 } } )$$

Evaluate

$$\frac{85}{88}\approx 0.965909091$$

Solution Steps

Calculate 11 to the power of 2 and get 121.

$$\left(1-\frac{1}{121}\right)\left(1-\frac{1}{12^{2}}\right)\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Convert 1 to fraction \frac{121}{121}.

$$\left(\frac{121}{121}-\frac{1}{121}\right)\left(1-\frac{1}{12^{2}}\right)\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Since \frac{121}{121} and \frac{1}{121} have the same denominator, subtract them by subtracting their numerators.

$$\frac{121-1}{121}\left(1-\frac{1}{12^{2}}\right)\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Subtract 1 from 121 to get 120.

$$\frac{120}{121}\left(1-\frac{1}{12^{2}}\right)\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Calculate 12 to the power of 2 and get 144.

$$\frac{120}{121}\left(1-\frac{1}{144}\right)\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Convert 1 to fraction \frac{144}{144}.

$$\frac{120}{121}\left(\frac{144}{144}-\frac{1}{144}\right)\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Since \frac{144}{144} and \frac{1}{144} have the same denominator, subtract them by subtracting their numerators.

$$\frac{120}{121}\times \frac{144-1}{144}\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Subtract 1 from 144 to get 143.

$$\frac{120}{121}\times \frac{143}{144}\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Multiply \frac{120}{121} times \frac{143}{144} by multiplying numerator times numerator and denominator times denominator.

$$\frac{120\times 143}{121\times 144}\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Do the multiplications in the fraction \frac{120\times 143}{121\times 144}.

$$\frac{17160}{17424}\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Reduce the fraction \frac{17160}{17424} to lowest terms by extracting and canceling out 264.

$$\frac{65}{66}\left(1-\frac{1}{13^{2}}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Calculate 13 to the power of 2 and get 169.

$$\frac{65}{66}\left(1-\frac{1}{169}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Convert 1 to fraction \frac{169}{169}.

$$\frac{65}{66}\left(\frac{169}{169}-\frac{1}{169}\right)\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Since \frac{169}{169} and \frac{1}{169} have the same denominator, subtract them by subtracting their numerators.

$$\frac{65}{66}\times \frac{169-1}{169}\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Subtract 1 from 169 to get 168.

$$\frac{65}{66}\times \frac{168}{169}\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Multiply \frac{65}{66} times \frac{168}{169} by multiplying numerator times numerator and denominator times denominator.

$$\frac{65\times 168}{66\times 169}\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Do the multiplications in the fraction \frac{65\times 168}{66\times 169}.

$$\frac{10920}{11154}\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Reduce the fraction \frac{10920}{11154} to lowest terms by extracting and canceling out 78.

$$\frac{140}{143}\left(1-\frac{1}{14^{2}}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Calculate 14 to the power of 2 and get 196.

$$\frac{140}{143}\left(1-\frac{1}{196}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Convert 1 to fraction \frac{196}{196}.

$$\frac{140}{143}\left(\frac{196}{196}-\frac{1}{196}\right)\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Since \frac{196}{196} and \frac{1}{196} have the same denominator, subtract them by subtracting their numerators.

$$\frac{140}{143}\times \frac{196-1}{196}\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Subtract 1 from 196 to get 195.

$$\frac{140}{143}\times \frac{195}{196}\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Multiply \frac{140}{143} times \frac{195}{196} by multiplying numerator times numerator and denominator times denominator.

$$\frac{140\times 195}{143\times 196}\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Do the multiplications in the fraction \frac{140\times 195}{143\times 196}.

$$\frac{27300}{28028}\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Reduce the fraction \frac{27300}{28028} to lowest terms by extracting and canceling out 364.

$$\frac{75}{77}\left(1-\frac{1}{15^{2}}\right)\left(1-\frac{1}{16^{2}}\right)$$

Calculate 15 to the power of 2 and get 225.

$$\frac{75}{77}\left(1-\frac{1}{225}\right)\left(1-\frac{1}{16^{2}}\right)$$

Convert 1 to fraction \frac{225}{225}.

$$\frac{75}{77}\left(\frac{225}{225}-\frac{1}{225}\right)\left(1-\frac{1}{16^{2}}\right)$$

Since \frac{225}{225} and \frac{1}{225} have the same denominator, subtract them by subtracting their numerators.

$$\frac{75}{77}\times \frac{225-1}{225}\left(1-\frac{1}{16^{2}}\right)$$

Subtract 1 from 225 to get 224.

$$\frac{75}{77}\times \frac{224}{225}\left(1-\frac{1}{16^{2}}\right)$$

Multiply \frac{75}{77} times \frac{224}{225} by multiplying numerator times numerator and denominator times denominator.

$$\frac{75\times 224}{77\times 225}\left(1-\frac{1}{16^{2}}\right)$$

Do the multiplications in the fraction \frac{75\times 224}{77\times 225}.

$$\frac{16800}{17325}\left(1-\frac{1}{16^{2}}\right)$$

Reduce the fraction \frac{16800}{17325} to lowest terms by extracting and canceling out 525.

$$\frac{32}{33}\left(1-\frac{1}{16^{2}}\right)$$

Calculate 16 to the power of 2 and get 256.

$$\frac{32}{33}\left(1-\frac{1}{256}\right)$$

Convert 1 to fraction \frac{256}{256}.

$$\frac{32}{33}\left(\frac{256}{256}-\frac{1}{256}\right)$$

Since \frac{256}{256} and \frac{1}{256} have the same denominator, subtract them by subtracting their numerators.

$$\frac{32}{33}\times \frac{256-1}{256}$$

Subtract 1 from 256 to get 255.

$$\frac{32}{33}\times \frac{255}{256}$$

Multiply \frac{32}{33} times \frac{255}{256} by multiplying numerator times numerator and denominator times denominator.

$$\frac{32\times 255}{33\times 256}$$

Do the multiplications in the fraction \frac{32\times 255}{33\times 256}.

$$\frac{8160}{8448}$$

Reduce the fraction \frac{8160}{8448} to lowest terms by extracting and canceling out 96.

$$\frac{85}{88}$$

Factor

$$\frac{5 \cdot 17}{2 ^ {3} \cdot 11} = 0.9659090909090909$$