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

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

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

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

Fraction \frac{-1}{9} can be rewritten as -\frac{1}{9} by extracting the negative sign.

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

Multiply -1 and -1 to get 1.

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

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

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

Multiply -1 and -\frac{1}{3} to get \frac{1}{3}.

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

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

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

Combine -\frac{1}{3}ab and \frac{1}{3}ba to get 0.

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

Expand

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

Solution Steps

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

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

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

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

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

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

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

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

Fraction \frac{-1}{9} can be rewritten as -\frac{1}{9} by extracting the negative sign.

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

Multiply -1 and -1 to get 1.

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

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

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

Multiply -1 and -\frac{1}{3} to get \frac{1}{3}.

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

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

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

Combine -\frac{1}{3}ab and \frac{1}{3}ba to get 0.

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