Solve {l}{-[-(1/2xy^{2})]-(-4x^{2})-4/3y^{4}-{[-(-1/3xy^{2})-1/2y^{4}-(-frac{1}{6}xy^{2})+(-frac{1}{6}y^{4})]}}{-a^{2}+(frac{1}{5}a^{3}b^{2}):(-frac{2}{5}ab^2)]:(-6a)+[(-2ab)^4:(+2ab^2)^2]:(+4a)}

$$\left. \begin{array} { l } { - [ - ( \frac { 1 } { 2 } x y ^ { 2 } ) ] - ( - 4 x ^ { 2 } ) - \frac { 4 } { 3 } y ^ { 4 } - \{ [ - ( - \frac { 1 } { 3 } x y ^ { 2 } ) - \frac { 1 } { 2 } y ^ { 4 } - ( - \frac { 1 } { 6 } x y ^ { 2 } ) + ( - \frac { 1 } { 6 } y ^ { 4 } ) ] \} } \\ { - a ^ { 2 } + ( \frac { 1 } { 5 } a ^ { 3 } b ^ { 2 } ) : ( - \frac { 2 } { 5 } a b ^ { 2 } ) ] : ( - 6 a ) + [ ( - 2 a b ) ^ { 4 } : ( + 2 a b ^ { 2 } ) ^ { 2 } ] : ( + 4 a ) } \end{array} \right.$$

Evaluate

$$-\frac{2y^{4}}{3}-\frac{3a^{2}}{2}+4x^{2}$$

Solution Steps

The opposite of -4x^{2} is 4x^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(-\left(-\frac{1}{3}xy^{2}\right)-\frac{1}{2}y^{4}-\left(-\frac{1}{6}xy^{2}\right)-\frac{1}{6}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

The opposite of -\frac{1}{3}xy^{2} is \frac{1}{3}xy^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(\frac{1}{3}xy^{2}-\frac{1}{2}y^{4}-\left(-\frac{1}{6}xy^{2}\right)-\frac{1}{6}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

The opposite of -\frac{1}{6}xy^{2} is \frac{1}{6}xy^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(\frac{1}{3}xy^{2}-\frac{1}{2}y^{4}+\frac{1}{6}xy^{2}-\frac{1}{6}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

Combine \frac{1}{3}xy^{2} and \frac{1}{6}xy^{2} to get \frac{1}{2}xy^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(\frac{1}{2}xy^{2}-\frac{1}{2}y^{4}-\frac{1}{6}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

Combine -\frac{1}{2}y^{4} and -\frac{1}{6}y^{4} to get -\frac{2}{3}y^{4}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(\frac{1}{2}xy^{2}-\frac{2}{3}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

To find the opposite of \frac{1}{2}xy^{2}-\frac{2}{3}y^{4}, find the opposite of each term.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\frac{1}{2}xy^{2}+\frac{2}{3}y^{4}-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

Combine -\frac{4}{3}y^{4} and \frac{2}{3}y^{4} to get -\frac{2}{3}y^{4}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

Cancel out ab^{2} in both numerator and denominator.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-a^{2}+\frac{\frac{1}{5}a^{2}}{-\frac{2}{5}}$$

Divide \frac{1}{5}a^{2} by -\frac{2}{5} by multiplying \frac{1}{5}a^{2} by the reciprocal of -\frac{2}{5}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-a^{2}+\frac{\frac{1}{5}a^{2}\times 5}{-2}$$

Multiply \frac{1}{5} and 5 to get 1.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-a^{2}+\frac{a^{2}}{-2}$$

Combine -a^{2} and \frac{a^{2}}{-2} to get -\frac{3}{2}a^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-\frac{3}{2}a^{2}$$

Multiply -1 and -1 to get 1.

$$\frac{1}{2}xy^{2}+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-\frac{3}{2}a^{2}$$

Combine \frac{1}{2}xy^{2} and -\frac{1}{2}xy^{2} to get 0.

$$4x^{2}-\frac{2}{3}y^{4}-\frac{3}{2}a^{2}$$

Expand

$$-\frac{2y^{4}}{3}-\frac{3a^{2}}{2}+4x^{2}$$

Solution Steps

The opposite of -4x^{2} is 4x^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(-\left(-\frac{1}{3}xy^{2}\right)-\frac{1}{2}y^{4}-\left(-\frac{1}{6}xy^{2}\right)-\frac{1}{6}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

The opposite of -\frac{1}{3}xy^{2} is \frac{1}{3}xy^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(\frac{1}{3}xy^{2}-\frac{1}{2}y^{4}-\left(-\frac{1}{6}xy^{2}\right)-\frac{1}{6}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

The opposite of -\frac{1}{6}xy^{2} is \frac{1}{6}xy^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(\frac{1}{3}xy^{2}-\frac{1}{2}y^{4}+\frac{1}{6}xy^{2}-\frac{1}{6}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

Combine \frac{1}{3}xy^{2} and \frac{1}{6}xy^{2} to get \frac{1}{2}xy^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(\frac{1}{2}xy^{2}-\frac{1}{2}y^{4}-\frac{1}{6}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

Combine -\frac{1}{2}y^{4} and -\frac{1}{6}y^{4} to get -\frac{2}{3}y^{4}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\left(\frac{1}{2}xy^{2}-\frac{2}{3}y^{4}\right)-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

To find the opposite of \frac{1}{2}xy^{2}-\frac{2}{3}y^{4}, find the opposite of each term.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{4}{3}y^{4}-\frac{1}{2}xy^{2}+\frac{2}{3}y^{4}-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

Combine -\frac{4}{3}y^{4} and \frac{2}{3}y^{4} to get -\frac{2}{3}y^{4}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-a^{2}+\frac{\frac{1}{5}a^{3}b^{2}}{-\frac{2}{5}ab^{2}}$$

Cancel out ab^{2} in both numerator and denominator.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-a^{2}+\frac{\frac{1}{5}a^{2}}{-\frac{2}{5}}$$

Divide \frac{1}{5}a^{2} by -\frac{2}{5} by multiplying \frac{1}{5}a^{2} by the reciprocal of -\frac{2}{5}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-a^{2}+\frac{\frac{1}{5}a^{2}\times 5}{-2}$$

Multiply \frac{1}{5} and 5 to get 1.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-a^{2}+\frac{a^{2}}{-2}$$

Combine -a^{2} and \frac{a^{2}}{-2} to get -\frac{3}{2}a^{2}.

$$-\left(-\frac{1}{2}xy^{2}\right)+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-\frac{3}{2}a^{2}$$

Multiply -1 and -1 to get 1.

$$\frac{1}{2}xy^{2}+4x^{2}-\frac{2}{3}y^{4}-\frac{1}{2}xy^{2}-\frac{3}{2}a^{2}$$

Combine \frac{1}{2}xy^{2} and -\frac{1}{2}xy^{2} to get 0.

$$4x^{2}-\frac{2}{3}y^{4}-\frac{3}{2}a^{2}$$