Solve left(1/2a-2/3bright)left(2/3a+3/4bright)

$$\left( \frac{ 1 }{ 2 } a- \frac{ 2 }{ 3 } b \right) \left( \frac{ 2 }{ 3 } a+ \frac{ 3 }{ 4 } b \right)$$

Evaluate

$$-\frac{5ab}{72}+\frac{a^{2}}{3}-\frac{b^{2}}{2}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{1}{2}a-\frac{2}{3}b by each term of \frac{2}{3}a+\frac{3}{4}b.

$$\frac{1}{2}a\times \frac{2}{3}a+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b\times \frac{3}{4}b$$

Multiply a and a to get a^{2}.

$$\frac{1}{2}a^{2}\times \frac{2}{3}+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b\times \frac{3}{4}b$$

Multiply b and b to get b^{2}.

$$\frac{1}{2}a^{2}\times \frac{2}{3}+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Multiply \frac{1}{2} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2}{2\times 3}a^{2}+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Cancel out 2 in both numerator and denominator.

$$\frac{1}{3}a^{2}+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Multiply \frac{1}{2} times \frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{3}a^{2}+\frac{1\times 3}{2\times 4}ab-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Do the multiplications in the fraction \frac{1\times 3}{2\times 4}.

$$\frac{1}{3}a^{2}+\frac{3}{8}ab-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

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

$$\frac{1}{3}a^{2}+\frac{3}{8}ab+\frac{-2\times 2}{3\times 3}ba-\frac{2}{3}b^{2}\times \frac{3}{4}$$

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

$$\frac{1}{3}a^{2}+\frac{3}{8}ab+\frac{-4}{9}ba-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Fraction \frac{-4}{9} can be rewritten as -\frac{4}{9} by extracting the negative sign.

$$\frac{1}{3}a^{2}+\frac{3}{8}ab-\frac{4}{9}ba-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Combine \frac{3}{8}ab and -\frac{4}{9}ba to get -\frac{5}{72}ab.

$$\frac{1}{3}a^{2}-\frac{5}{72}ab-\frac{2}{3}b^{2}\times \frac{3}{4}$$

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

$$\frac{1}{3}a^{2}-\frac{5}{72}ab+\frac{-2\times 3}{3\times 4}b^{2}$$

Cancel out 3 in both numerator and denominator.

$$\frac{1}{3}a^{2}-\frac{5}{72}ab+\frac{-2}{4}b^{2}$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{1}{3}a^{2}-\frac{5}{72}ab-\frac{1}{2}b^{2}$$

Expand

$$-\frac{5ab}{72}+\frac{a^{2}}{3}-\frac{b^{2}}{2}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{1}{2}a-\frac{2}{3}b by each term of \frac{2}{3}a+\frac{3}{4}b.

$$\frac{1}{2}a\times \frac{2}{3}a+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b\times \frac{3}{4}b$$

Multiply a and a to get a^{2}.

$$\frac{1}{2}a^{2}\times \frac{2}{3}+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b\times \frac{3}{4}b$$

Multiply b and b to get b^{2}.

$$\frac{1}{2}a^{2}\times \frac{2}{3}+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Multiply \frac{1}{2} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2}{2\times 3}a^{2}+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Cancel out 2 in both numerator and denominator.

$$\frac{1}{3}a^{2}+\frac{1}{2}a\times \frac{3}{4}b-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Multiply \frac{1}{2} times \frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{3}a^{2}+\frac{1\times 3}{2\times 4}ab-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Do the multiplications in the fraction \frac{1\times 3}{2\times 4}.

$$\frac{1}{3}a^{2}+\frac{3}{8}ab-\frac{2}{3}b\times \frac{2}{3}a-\frac{2}{3}b^{2}\times \frac{3}{4}$$

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

$$\frac{1}{3}a^{2}+\frac{3}{8}ab+\frac{-2\times 2}{3\times 3}ba-\frac{2}{3}b^{2}\times \frac{3}{4}$$

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

$$\frac{1}{3}a^{2}+\frac{3}{8}ab+\frac{-4}{9}ba-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Fraction \frac{-4}{9} can be rewritten as -\frac{4}{9} by extracting the negative sign.

$$\frac{1}{3}a^{2}+\frac{3}{8}ab-\frac{4}{9}ba-\frac{2}{3}b^{2}\times \frac{3}{4}$$

Combine \frac{3}{8}ab and -\frac{4}{9}ba to get -\frac{5}{72}ab.

$$\frac{1}{3}a^{2}-\frac{5}{72}ab-\frac{2}{3}b^{2}\times \frac{3}{4}$$

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

$$\frac{1}{3}a^{2}-\frac{5}{72}ab+\frac{-2\times 3}{3\times 4}b^{2}$$

Cancel out 3 in both numerator and denominator.

$$\frac{1}{3}a^{2}-\frac{5}{72}ab+\frac{-2}{4}b^{2}$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{1}{3}a^{2}-\frac{5}{72}ab-\frac{1}{2}b^{2}$$