Solve (2m/3+3n/4)(-3n/4+2m/3)

$$( \frac { 2 m } { 3 } + \frac { 3 n } { 4 } ) ( - \frac { 3 n } { 4 } + \frac { 2 m } { 3 } )$$

Evaluate

$$\frac{4m^{2}}{9}-\frac{9n^{2}}{16}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 4 is 12. Multiply \frac{2m}{3} times \frac{4}{4}. Multiply \frac{3n}{4} times \frac{3}{3}.

$$\left(\frac{4\times 2m}{12}+\frac{3\times 3n}{12}\right)\left(-\frac{3n}{4}+\frac{2m}{3}\right)$$

Since \frac{4\times 2m}{12} and \frac{3\times 3n}{12} have the same denominator, add them by adding their numerators.

$$\frac{4\times 2m+3\times 3n}{12}\left(-\frac{3n}{4}+\frac{2m}{3}\right)$$

Do the multiplications in 4\times 2m+3\times 3n.

$$\frac{8m+9n}{12}\left(-\frac{3n}{4}+\frac{2m}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 3 is 12. Multiply -\frac{3n}{4} times \frac{3}{3}. Multiply \frac{2m}{3} times \frac{4}{4}.

$$\frac{8m+9n}{12}\left(-\frac{3\times 3n}{12}+\frac{4\times 2m}{12}\right)$$

Since -\frac{3\times 3n}{12} and \frac{4\times 2m}{12} have the same denominator, add them by adding their numerators.

$$\frac{8m+9n}{12}\times \frac{-3\times 3n+4\times 2m}{12}$$

Do the multiplications in -3\times 3n+4\times 2m.

$$\frac{8m+9n}{12}\times \frac{-9n+8m}{12}$$

Multiply \frac{8m+9n}{12} times \frac{-9n+8m}{12} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(8m+9n\right)\left(-9n+8m\right)}{12\times 12}$$

Multiply 12 and 12 to get 144.

$$\frac{\left(8m+9n\right)\left(-9n+8m\right)}{144}$$

Apply the distributive property by multiplying each term of 8m+9n by each term of -9n+8m.

$$\frac{-72mn+64m^{2}-81n^{2}+72nm}{144}$$

Combine -72mn and 72nm to get 0.

$$\frac{64m^{2}-81n^{2}}{144}$$

Expand

$$\frac{4m^{2}}{9}-\frac{9n^{2}}{16}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 4 is 12. Multiply \frac{2m}{3} times \frac{4}{4}. Multiply \frac{3n}{4} times \frac{3}{3}.

$$\left(\frac{4\times 2m}{12}+\frac{3\times 3n}{12}\right)\left(-\frac{3n}{4}+\frac{2m}{3}\right)$$

Since \frac{4\times 2m}{12} and \frac{3\times 3n}{12} have the same denominator, add them by adding their numerators.

$$\frac{4\times 2m+3\times 3n}{12}\left(-\frac{3n}{4}+\frac{2m}{3}\right)$$

Do the multiplications in 4\times 2m+3\times 3n.

$$\frac{8m+9n}{12}\left(-\frac{3n}{4}+\frac{2m}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 3 is 12. Multiply -\frac{3n}{4} times \frac{3}{3}. Multiply \frac{2m}{3} times \frac{4}{4}.

$$\frac{8m+9n}{12}\left(-\frac{3\times 3n}{12}+\frac{4\times 2m}{12}\right)$$

Since -\frac{3\times 3n}{12} and \frac{4\times 2m}{12} have the same denominator, add them by adding their numerators.

$$\frac{8m+9n}{12}\times \frac{-3\times 3n+4\times 2m}{12}$$

Do the multiplications in -3\times 3n+4\times 2m.

$$\frac{8m+9n}{12}\times \frac{-9n+8m}{12}$$

Multiply \frac{8m+9n}{12} times \frac{-9n+8m}{12} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(8m+9n\right)\left(-9n+8m\right)}{12\times 12}$$

Multiply 12 and 12 to get 144.

$$\frac{\left(8m+9n\right)\left(-9n+8m\right)}{144}$$

Apply the distributive property by multiplying each term of 8m+9n by each term of -9n+8m.

$$\frac{-72mn+64m^{2}-81n^{2}+72nm}{144}$$

Combine -72mn and 72nm to get 0.

$$\frac{64m^{2}-81n^{2}}{144}$$