Solve (1/3+x/2)(1/9-x^2/4)(frac{1}{3}-frac{x}{2})

$$( \frac { 1 } { 3 } + \frac { x } { 2 } ) ( \frac { 1 } { 9 } - \frac { x ^ { 2 } } { 4 } ) ( \frac { 1 } { 3 } - \frac { x } { 2 } )$$

Evaluate

$$\frac{\left(4-9x^{2}\right)^{2}}{1296}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{1}{3} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{3}{3}.

$$\left(\frac{2}{6}+\frac{3x}{6}\right)\left(\frac{1}{9}-\frac{x^{2}}{4}\right)\left(\frac{1}{3}-\frac{x}{2}\right)$$

Since \frac{2}{6} and \frac{3x}{6} have the same denominator, add them by adding their numerators.

$$\frac{2+3x}{6}\left(\frac{1}{9}-\frac{x^{2}}{4}\right)\left(\frac{1}{3}-\frac{x}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 4 is 36. Multiply \frac{1}{9} times \frac{4}{4}. Multiply \frac{x^{2}}{4} times \frac{9}{9}.

$$\frac{2+3x}{6}\left(\frac{4}{36}-\frac{9x^{2}}{36}\right)\left(\frac{1}{3}-\frac{x}{2}\right)$$

Since \frac{4}{36} and \frac{9x^{2}}{36} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2+3x}{6}\times \frac{4-9x^{2}}{36}\left(\frac{1}{3}-\frac{x}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{1}{3} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{3}{3}.

$$\frac{2+3x}{6}\times \frac{4-9x^{2}}{36}\left(\frac{2}{6}-\frac{3x}{6}\right)$$

Since \frac{2}{6} and \frac{3x}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2+3x}{6}\times \frac{4-9x^{2}}{36}\times \frac{2-3x}{6}$$

Multiply \frac{2+3x}{6} times \frac{4-9x^{2}}{36} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2+3x\right)\left(4-9x^{2}\right)}{6\times 36}\times \frac{2-3x}{6}$$

Multiply \frac{\left(2+3x\right)\left(4-9x^{2}\right)}{6\times 36} times \frac{2-3x}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2+3x\right)\left(4-9x^{2}\right)\left(2-3x\right)}{6\times 36\times 6}$$

Multiply 6 and 36 to get 216.

$$\frac{\left(2+3x\right)\left(4-9x^{2}\right)\left(2-3x\right)}{216\times 6}$$

Multiply 216 and 6 to get 1296.

$$\frac{\left(2+3x\right)\left(4-9x^{2}\right)\left(2-3x\right)}{1296}$$

Use the distributive property to multiply 2+3x by 4-9x^{2}.

$$\frac{\left(8-18x^{2}+12x-27x^{3}\right)\left(2-3x\right)}{1296}$$

Use the distributive property to multiply 8-18x^{2}+12x-27x^{3} by 2-3x and combine like terms.

$$\frac{16-72x^{2}+81x^{4}}{1296}$$

Expand

$$\frac{x^{4}}{16}-\frac{x^{2}}{18}+\frac{1}{81}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{1}{3} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{3}{3}.

$$\left(\frac{2}{6}+\frac{3x}{6}\right)\left(\frac{1}{9}-\frac{x^{2}}{4}\right)\left(\frac{1}{3}-\frac{x}{2}\right)$$

Since \frac{2}{6} and \frac{3x}{6} have the same denominator, add them by adding their numerators.

$$\frac{2+3x}{6}\left(\frac{1}{9}-\frac{x^{2}}{4}\right)\left(\frac{1}{3}-\frac{x}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 4 is 36. Multiply \frac{1}{9} times \frac{4}{4}. Multiply \frac{x^{2}}{4} times \frac{9}{9}.

$$\frac{2+3x}{6}\left(\frac{4}{36}-\frac{9x^{2}}{36}\right)\left(\frac{1}{3}-\frac{x}{2}\right)$$

Since \frac{4}{36} and \frac{9x^{2}}{36} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2+3x}{6}\times \frac{4-9x^{2}}{36}\left(\frac{1}{3}-\frac{x}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{1}{3} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{3}{3}.

$$\frac{2+3x}{6}\times \frac{4-9x^{2}}{36}\left(\frac{2}{6}-\frac{3x}{6}\right)$$

Since \frac{2}{6} and \frac{3x}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2+3x}{6}\times \frac{4-9x^{2}}{36}\times \frac{2-3x}{6}$$

Multiply \frac{2+3x}{6} times \frac{4-9x^{2}}{36} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2+3x\right)\left(4-9x^{2}\right)}{6\times 36}\times \frac{2-3x}{6}$$

Multiply \frac{\left(2+3x\right)\left(4-9x^{2}\right)}{6\times 36} times \frac{2-3x}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2+3x\right)\left(4-9x^{2}\right)\left(2-3x\right)}{6\times 36\times 6}$$

Multiply 6 and 36 to get 216.

$$\frac{\left(2+3x\right)\left(4-9x^{2}\right)\left(2-3x\right)}{216\times 6}$$

Multiply 216 and 6 to get 1296.

$$\frac{\left(2+3x\right)\left(4-9x^{2}\right)\left(2-3x\right)}{1296}$$

Use the distributive property to multiply 2+3x by 4-9x^{2}.

$$\frac{\left(8-18x^{2}+12x-27x^{3}\right)\left(2-3x\right)}{1296}$$

Use the distributive property to multiply 8-18x^{2}+12x-27x^{3} by 2-3x and combine like terms.

$$\frac{16-72x^{2}+81x^{4}}{1296}$$