Problem 76 - College Algebra

Question

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

Step-by-Step Solution

Answer

The product of the given expression \((x^{2} y^{2} - 3)^2 \)is \( x^{4} y^{4} - 6x^{2} y^{2} + 9 \)

Step 1: Identifying the Elements

Identify \(a\) and \(b\) from the given expression. Here, \(a = x^{2} y^{2}\) and \(b = 3\).

Step 2: Applying the Binomial Expansion

Apply the binomial expansion formula \((a - b)^2 = a^2 - 2ab + b^2 \). Substitute \(a\) and \(b\) with their respective values that we have identified in step 1. So, \((x^{2} y^{2} - 3)^2 = (x^{2} y^{2})^2 - 2 * x^{2} y^{2} * 3 + 3^2\)

Step 3: Simplify the Equation

It's time for simplification. The square power \( (x^{2} y^{2})^2 \) becomes \(x^{4} y^{4}\), the multiplication \(2 * x^{2} y^{2} * 3\) equals \(6x^{2} y^{2}\), and \(3^2\) equals 9. So, the simplified form of the given expression is \(x^{4} y^{4} - 6x^{2} y^{2} + 9\)