Solve (1/2a+1/3b)(3/2a-2/3b)

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

Evaluate

$$\frac{ab}{6}+\frac{3a^{2}}{4}-\frac{2b^{2}}{9}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Do the multiplications in the fraction \frac{1\left(-2\right)}{2\times 3}.

$$\frac{3}{4}a^{2}+\frac{-2}{6}ab+\frac{1}{3}b\times \frac{3}{2}a+\frac{1}{3}b^{2}\left(-\frac{2}{3}\right)$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

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

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

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

Cancel out 3 in both numerator and denominator.

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

Combine -\frac{1}{3}ab and \frac{1}{2}ba to get \frac{1}{6}ab.

$$\frac{3}{4}a^{2}+\frac{1}{6}ab+\frac{1}{3}b^{2}\left(-\frac{2}{3}\right)$$

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

$$\frac{3}{4}a^{2}+\frac{1}{6}ab+\frac{1\left(-2\right)}{3\times 3}b^{2}$$

Do the multiplications in the fraction \frac{1\left(-2\right)}{3\times 3}.

$$\frac{3}{4}a^{2}+\frac{1}{6}ab+\frac{-2}{9}b^{2}$$

Fraction \frac{-2}{9} can be rewritten as -\frac{2}{9} by extracting the negative sign.

$$\frac{3}{4}a^{2}+\frac{1}{6}ab-\frac{2}{9}b^{2}$$

Expand

$$\frac{ab}{6}+\frac{3a^{2}}{4}-\frac{2b^{2}}{9}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Do the multiplications in the fraction \frac{1\left(-2\right)}{2\times 3}.

$$\frac{3}{4}a^{2}+\frac{-2}{6}ab+\frac{1}{3}b\times \frac{3}{2}a+\frac{1}{3}b^{2}\left(-\frac{2}{3}\right)$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

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

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

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

Cancel out 3 in both numerator and denominator.

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

Combine -\frac{1}{3}ab and \frac{1}{2}ba to get \frac{1}{6}ab.

$$\frac{3}{4}a^{2}+\frac{1}{6}ab+\frac{1}{3}b^{2}\left(-\frac{2}{3}\right)$$

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

$$\frac{3}{4}a^{2}+\frac{1}{6}ab+\frac{1\left(-2\right)}{3\times 3}b^{2}$$

Do the multiplications in the fraction \frac{1\left(-2\right)}{3\times 3}.

$$\frac{3}{4}a^{2}+\frac{1}{6}ab+\frac{-2}{9}b^{2}$$

Fraction \frac{-2}{9} can be rewritten as -\frac{2}{9} by extracting the negative sign.

$$\frac{3}{4}a^{2}+\frac{1}{6}ab-\frac{2}{9}b^{2}$$