(1/2a-3b)(3b+1/2a)(1/4a^2+9b^2)

Jasonbradley

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Jun 16, 2024

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Expanding the Expression: (1/2a - 3b)(3b + 1/2a)(1/4a^2 + 9b^2)

This expression involves multiplying three binomials together, which can be done in a systematic way using the distributive property or by recognizing certain patterns. Let's break down the steps:

Step 1: Multiply the first two binomials

We have (1/2a - 3b)(3b + 1/2a). This resembles the pattern of a difference of squares: (x - y)(x + y) = x² - y².

Let's apply this pattern:

• x = 1/2a
• y = 3b

Therefore: (1/2a - 3b)(3b + 1/2a) = (1/2a)² - (3b)² = 1/4a² - 9b²

Step 2: Multiply the result from Step 1 by the remaining binomial

Now we have (1/4a² - 9b²)(1/4a² + 9b²). Again, this resembles the difference of squares pattern:

• x = 1/4a²
• y = 9b²

Therefore: (1/4a² - 9b²)(1/4a² + 9b²) = (1/4a²)² - (9b²)² = 1/16a⁴ - 81b⁴

Final Result

The expanded form of the expression (1/2a - 3b)(3b + 1/2a)(1/4a² + 9b²) is 1/16a⁴ - 81b⁴.