Solve a^4b^4-1

$$a ^ { 4 } b ^ { 4 } - 1$$

Factor

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

Solution Steps

Rewrite a^{4}b^{4}-1 as \left(a^{2}b^{2}\right)^{2}-1^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

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

Consider a^{2}b^{2}-1. Rewrite a^{2}b^{2}-1 as \left(ab\right)^{2}-1^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

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

Rewrite the complete factored expression.

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

Evaluate

$$\left(ab\right)^{4}-1$$