Solve (1/2xy-y^{2})*(1/2xy+y^{2})-y^2*(y^2+1/4x^2)+4y^2=

$$( \frac { 1 } { 2 } x y - y ^ { 2 } ) \cdot ( \frac { 1 } { 2 } x y + y ^ { 2 } ) - y ^ { 2 } \cdot ( y ^ { 2 } + \frac { 1 } { 4 } x ^ { 2 } ) + 4 y ^ { 2 } =$$

Evaluate

$$4y^{2}-2y^{4}$$

Solution Steps

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

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

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

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

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

Use the distributive property to multiply y^{2} by y^{2}+\frac{1}{4}x^{2}.

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

To find the opposite of y^{4}+\frac{1}{4}y^{2}x^{2}, find the opposite of each term.

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

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

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

Combine \frac{1}{4}x^{2}y^{2} and -\frac{1}{4}y^{2}x^{2} to get 0.

$$-2y^{4}+4y^{2}$$

Expand

$$4y^{2}-2y^{4}$$

Solution Steps

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

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

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

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

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

Use the distributive property to multiply y^{2} by y^{2}+\frac{1}{4}x^{2}.

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

To find the opposite of y^{4}+\frac{1}{4}y^{2}x^{2}, find the opposite of each term.

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

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

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

Combine \frac{1}{4}x^{2}y^{2} and -\frac{1}{4}y^{2}x^{2} to get 0.

$$-2y^{4}+4y^{2}$$