Solve (2/3a-1/2b)(2/3a+1/2b)-frac{1}{2}(a-frac{2}{3}b)+frac{2}{3}(b-frac{1}{2}a)

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

Evaluate

$$\frac{4a^{2}}{9}-\frac{b^{2}}{4}-\frac{5a}{6}+b$$

Solution Steps

Consider \left(\frac{2}{3}a-\frac{1}{2}b\right)\left(\frac{2}{3}a+\frac{1}{2}b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(\frac{2}{3}a\right)^{2}.

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

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

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

Expand \left(\frac{1}{2}b\right)^{2}.

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

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

Use the distributive property to multiply -\frac{1}{2} by a-\frac{2}{3}b.

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

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

$$\frac{4}{9}a^{2}-\frac{1}{4}b^{2}-\frac{1}{2}a+\frac{-\left(-2\right)}{2\times 3}b+\frac{2}{3}\left(b-\frac{1}{2}a\right)$$

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

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

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

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

Use the distributive property to multiply \frac{2}{3} by b-\frac{1}{2}a.

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

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

$$\frac{4}{9}a^{2}-\frac{1}{4}b^{2}-\frac{1}{2}a+\frac{1}{3}b+\frac{2}{3}b+\frac{2\left(-1\right)}{3\times 2}a$$

Cancel out 2 in both numerator and denominator.

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

Fraction \frac{-1}{3} can be rewritten as -\frac{1}{3} by extracting the negative sign.

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

Combine \frac{1}{3}b and \frac{2}{3}b to get b.

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

Combine -\frac{1}{2}a and -\frac{1}{3}a to get -\frac{5}{6}a.

$$\frac{4}{9}a^{2}-\frac{1}{4}b^{2}-\frac{5}{6}a+b$$

Expand

$$\frac{4a^{2}}{9}-\frac{b^{2}}{4}-\frac{5a}{6}+b$$

Solution Steps

Consider \left(\frac{2}{3}a-\frac{1}{2}b\right)\left(\frac{2}{3}a+\frac{1}{2}b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(\frac{2}{3}a\right)^{2}.

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

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

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

Expand \left(\frac{1}{2}b\right)^{2}.

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

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

Use the distributive property to multiply -\frac{1}{2} by a-\frac{2}{3}b.

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

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

$$\frac{4}{9}a^{2}-\frac{1}{4}b^{2}-\frac{1}{2}a+\frac{-\left(-2\right)}{2\times 3}b+\frac{2}{3}\left(b-\frac{1}{2}a\right)$$

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

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

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

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

Use the distributive property to multiply \frac{2}{3} by b-\frac{1}{2}a.

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

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

$$\frac{4}{9}a^{2}-\frac{1}{4}b^{2}-\frac{1}{2}a+\frac{1}{3}b+\frac{2}{3}b+\frac{2\left(-1\right)}{3\times 2}a$$

Cancel out 2 in both numerator and denominator.

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

Fraction \frac{-1}{3} can be rewritten as -\frac{1}{3} by extracting the negative sign.

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

Combine \frac{1}{3}b and \frac{2}{3}b to get b.

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

Combine -\frac{1}{2}a and -\frac{1}{3}a to get -\frac{5}{6}a.

$$\frac{4}{9}a^{2}-\frac{1}{4}b^{2}-\frac{5}{6}a+b$$