Problem 75 - College Algebra

Question

Find each product. $$\left(x^{2} y^{2}-3\right)^{2}$$

Step-by-Step Solution

Answer

The product is \(x^{4}y^{4}-6x^{2}y^{2}+9\)

Step 1: Understanding Binomial Squares

A binomial square is a binomial expression raised to the power of 2. The general formula for expanding binomial squares is \((a+b)^{2}=a^{2}+2ab+b^{2}\). In this case, \(a\) is \(x^{2}y^{2}\) and \(b\) is \(-3\), and we need to substitute these values into our formula.

Step 2: Applying the formula

Substituting \(a=x^{2}y^{2}\) and \(b=-3\) into the formula, we get: \((x^{2}y^{2}-3)^{2} = (x^{2}y^{2})^{2} + 2(x^{2}y^{2})(-3) + (-3)^{2}\)

Step 3: Solving for Each Term

Solving for each term we get: \((x^{2}y^{2})^{2} = x^{4}y^{4}\), \(2(x^{2}y^{2})(-3) = -6x^{2}y^{2}\) and \((-3)^{2} = 9\)

Step 4: Combining the Terms

Combining these three terms, the expansion of \((x^{2}y^{2}-3)^{2}\) is \(x^{4}y^{4}-6x^{2}y^{2}+9\)