Solve (3/4ab+1)(3/4ab-1)-(-3/4ab)^2+(1+ab)(1-ab)

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

Evaluate

$$-\left(ab\right)^{2}$$

Solution Steps

Consider \left(\frac{3}{4}ab+1\right)\left(\frac{3}{4}ab-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Expand \left(\frac{3}{4}ab\right)^{2}.

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

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

$$\frac{9}{16}a^{2}b^{2}-1-\left(-\frac{3}{4}ab\right)^{2}+\left(1+ab\right)\left(1-ab\right)$$

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

$$\frac{9}{16}a^{2}b^{2}-1-\left(-\frac{3}{4}\right)^{2}a^{2}b^{2}+\left(1+ab\right)\left(1-ab\right)$$

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

$$\frac{9}{16}a^{2}b^{2}-1-\frac{9}{16}a^{2}b^{2}+\left(1+ab\right)\left(1-ab\right)$$

Combine \frac{9}{16}a^{2}b^{2} and -\frac{9}{16}a^{2}b^{2} to get 0.

$$-1+\left(1+ab\right)\left(1-ab\right)$$

Use the distributive property to multiply 1+ab by 1-ab and combine like terms.

$$-1+1-a^{2}b^{2}$$

Add -1 and 1 to get 0.

$$-a^{2}b^{2}$$

Expand

$$-\left(ab\right)^{2}$$

Solution Steps

Consider \left(\frac{3}{4}ab+1\right)\left(\frac{3}{4}ab-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Expand \left(\frac{3}{4}ab\right)^{2}.

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

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

$$\frac{9}{16}a^{2}b^{2}-1-\left(-\frac{3}{4}ab\right)^{2}+\left(1+ab\right)\left(1-ab\right)$$

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

$$\frac{9}{16}a^{2}b^{2}-1-\left(-\frac{3}{4}\right)^{2}a^{2}b^{2}+\left(1+ab\right)\left(1-ab\right)$$

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

$$\frac{9}{16}a^{2}b^{2}-1-\frac{9}{16}a^{2}b^{2}+\left(1+ab\right)\left(1-ab\right)$$

Combine \frac{9}{16}a^{2}b^{2} and -\frac{9}{16}a^{2}b^{2} to get 0.

$$-1+\left(1+ab\right)\left(1-ab\right)$$

Use the distributive property to multiply 1+ab by 1-ab and combine like terms.

$$-1+1-a^{2}b^{2}$$

Add -1 and 1 to get 0.

$$-a^{2}b^{2}$$