Solve (-3)[(1/4a^{2}x^{2})^{2}-(1/2a^2x^2)^2]-{[3ax(-1/4ax)]^2-12}

$$( - 3 ) [ ( \frac { 1 } { 4 } a ^ { 2 } x ^ { 2 } ) ^ { 2 } - ( \frac { 1 } { 2 } a ^ { 2 } x ^ { 2 } ) ^ { 2 } ] - \{ [ 3 a x ( - \frac { 1 } { 4 } a x ) ] ^ { 2 } - 12 \}$$

Evaluate

$$12$$

Solution Steps

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

$$-3\left(\left(\frac{1}{4}a^{2}x^{2}\right)^{2}-\left(\frac{1}{2}a^{2}x^{2}\right)^{2}\right)-\left(\left(3a^{2}x\left(-\frac{1}{4}\right)x\right)^{2}-12\right)$$

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

$$-3\left(\left(\frac{1}{4}a^{2}x^{2}\right)^{2}-\left(\frac{1}{2}a^{2}x^{2}\right)^{2}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

Expand \left(\frac{1}{4}a^{2}x^{2}\right)^{2}.

$$-3\left(\left(\frac{1}{4}\right)^{2}\left(a^{2}\right)^{2}\left(x^{2}\right)^{2}-\left(\frac{1}{2}a^{2}x^{2}\right)^{2}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

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

$$-3\left(\left(\frac{1}{4}\right)^{2}a^{4}\left(x^{2}\right)^{2}-\left(\frac{1}{2}a^{2}x^{2}\right)^{2}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

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

$$-3\left(\left(\frac{1}{4}\right)^{2}a^{4}x^{4}-\left(\frac{1}{2}a^{2}x^{2}\right)^{2}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

Calculate \frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$-3\left(\frac{1}{16}a^{4}x^{4}-\left(\frac{1}{2}a^{2}x^{2}\right)^{2}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

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

$$-3\left(\frac{1}{16}a^{4}x^{4}-\left(\frac{1}{2}\right)^{2}\left(a^{2}\right)^{2}\left(x^{2}\right)^{2}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

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

$$-3\left(\frac{1}{16}a^{4}x^{4}-\left(\frac{1}{2}\right)^{2}a^{4}\left(x^{2}\right)^{2}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

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

$$-3\left(\frac{1}{16}a^{4}x^{4}-\left(\frac{1}{2}\right)^{2}a^{4}x^{4}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

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

$$-3\left(\frac{1}{16}a^{4}x^{4}-\frac{1}{4}a^{4}x^{4}\right)-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

Combine \frac{1}{16}a^{4}x^{4} and -\frac{1}{4}a^{4}x^{4} to get -\frac{3}{16}a^{4}x^{4}.

$$-3\left(-\frac{3}{16}\right)a^{4}x^{4}-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

Multiply -3 and -\frac{3}{16} to get \frac{9}{16}.

$$\frac{9}{16}a^{4}x^{4}-\left(\left(3a^{2}x^{2}\left(-\frac{1}{4}\right)\right)^{2}-12\right)$$

Multiply 3 and -\frac{1}{4} to get -\frac{3}{4}.

$$\frac{9}{16}a^{4}x^{4}-\left(\left(-\frac{3}{4}a^{2}x^{2}\right)^{2}-12\right)$$

Expand \left(-\frac{3}{4}a^{2}x^{2}\right)^{2}.

$$\frac{9}{16}a^{4}x^{4}-\left(\left(-\frac{3}{4}\right)^{2}\left(a^{2}\right)^{2}\left(x^{2}\right)^{2}-12\right)$$

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

$$\frac{9}{16}a^{4}x^{4}-\left(\left(-\frac{3}{4}\right)^{2}a^{4}\left(x^{2}\right)^{2}-12\right)$$

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

$$\frac{9}{16}a^{4}x^{4}-\left(\left(-\frac{3}{4}\right)^{2}a^{4}x^{4}-12\right)$$

Calculate -\frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$\frac{9}{16}a^{4}x^{4}-\left(\frac{9}{16}a^{4}x^{4}-12\right)$$

To find the opposite of \frac{9}{16}a^{4}x^{4}-12, find the opposite of each term.

$$\frac{9}{16}a^{4}x^{4}-\frac{9}{16}a^{4}x^{4}+12$$

Combine \frac{9}{16}a^{4}x^{4} and -\frac{9}{16}a^{4}x^{4} to get 0.

$$12$$

Factor

$$2^{2}\times 3$$

Solution Steps

Factor out \frac{3}{16}.

$$\frac{3\left(-\left(\left(a^{2}x^{2}\right)^{2}-4\left(a^{2}x^{2}\right)^{2}\right)-\left(3\left(axax\right)^{2}-64\right)\right)}{16}$$

Simplify.

$$12$$