Solve (1-1/3)(1-1/5)(1-1/7)/(1+frac{1{3})(1+frac{1}{5})(1+frac{1}{7})}

$$\frac { ( 1 - \frac { 1 } { 3 } ) ( 1 - \frac { 1 } { 5 } ) ( 1 - \frac { 1 } { 7 } ) } { ( 1 + \frac { 1 } { 3 } ) ( 1 + \frac { 1 } { 5 } ) ( 1 + \frac { 1 } { 7 } ) }$$

Evaluate

$$\frac{1}{4}=0.25$$

Solution Steps

Convert 1 to fraction \frac{3}{3}.

$$\frac{\left(\frac{3}{3}-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Since \frac{3}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3-1}{3}\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Subtract 1 from 3 to get 2.

$$\frac{\frac{2}{3}\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Convert 1 to fraction \frac{5}{5}.

$$\frac{\frac{2}{3}\left(\frac{5}{5}-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Since \frac{5}{5} and \frac{1}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2}{3}\times \frac{5-1}{5}\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Subtract 1 from 5 to get 4.

$$\frac{\frac{2}{3}\times \frac{4}{5}\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Multiply \frac{2}{3} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{2\times 4}{3\times 5}\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Do the multiplications in the fraction \frac{2\times 4}{3\times 5}.

$$\frac{\frac{8}{15}\left(1-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Convert 1 to fraction \frac{7}{7}.

$$\frac{\frac{8}{15}\left(\frac{7}{7}-\frac{1}{7}\right)}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Since \frac{7}{7} and \frac{1}{7} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8}{15}\times \frac{7-1}{7}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Subtract 1 from 7 to get 6.

$$\frac{\frac{8}{15}\times \frac{6}{7}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Multiply \frac{8}{15} times \frac{6}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{8\times 6}{15\times 7}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Do the multiplications in the fraction \frac{8\times 6}{15\times 7}.

$$\frac{\frac{48}{105}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Reduce the fraction \frac{48}{105} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{16}{35}}{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Convert 1 to fraction \frac{3}{3}.

$$\frac{\frac{16}{35}}{\left(\frac{3}{3}+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Since \frac{3}{3} and \frac{1}{3} have the same denominator, add them by adding their numerators.

$$\frac{\frac{16}{35}}{\frac{3+1}{3}\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Add 3 and 1 to get 4.

$$\frac{\frac{16}{35}}{\frac{4}{3}\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Convert 1 to fraction \frac{5}{5}.

$$\frac{\frac{16}{35}}{\frac{4}{3}\left(\frac{5}{5}+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$$

Since \frac{5}{5} and \frac{1}{5} have the same denominator, add them by adding their numerators.

$$\frac{\frac{16}{35}}{\frac{4}{3}\times \frac{5+1}{5}\left(1+\frac{1}{7}\right)}$$

Add 5 and 1 to get 6.

$$\frac{\frac{16}{35}}{\frac{4}{3}\times \frac{6}{5}\left(1+\frac{1}{7}\right)}$$

Multiply \frac{4}{3} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{16}{35}}{\frac{4\times 6}{3\times 5}\left(1+\frac{1}{7}\right)}$$

Do the multiplications in the fraction \frac{4\times 6}{3\times 5}.

$$\frac{\frac{16}{35}}{\frac{24}{15}\left(1+\frac{1}{7}\right)}$$

Reduce the fraction \frac{24}{15} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{16}{35}}{\frac{8}{5}\left(1+\frac{1}{7}\right)}$$

Convert 1 to fraction \frac{7}{7}.

$$\frac{\frac{16}{35}}{\frac{8}{5}\left(\frac{7}{7}+\frac{1}{7}\right)}$$

Since \frac{7}{7} and \frac{1}{7} have the same denominator, add them by adding their numerators.

$$\frac{\frac{16}{35}}{\frac{8}{5}\times \frac{7+1}{7}}$$

Add 7 and 1 to get 8.

$$\frac{\frac{16}{35}}{\frac{8}{5}\times \frac{8}{7}}$$

Multiply \frac{8}{5} times \frac{8}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{16}{35}}{\frac{8\times 8}{5\times 7}}$$

Do the multiplications in the fraction \frac{8\times 8}{5\times 7}.

$$\frac{\frac{16}{35}}{\frac{64}{35}}$$

Divide \frac{16}{35} by \frac{64}{35} by multiplying \frac{16}{35} by the reciprocal of \frac{64}{35}.

$$\frac{16}{35}\times \frac{35}{64}$$

Multiply \frac{16}{35} times \frac{35}{64} by multiplying numerator times numerator and denominator times denominator.

$$\frac{16\times 35}{35\times 64}$$

Cancel out 35 in both numerator and denominator.

$$\frac{16}{64}$$

Reduce the fraction \frac{16}{64} to lowest terms by extracting and canceling out 16.

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

Factor

$$\frac{1}{2 ^ {2}} = 0.25$$