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Chapter 5: Problem 52

Squares of Binomials Multiply. $$\left(a^{4} b^{2}-3\right)^{2}$$

Short Answer

Expert verified
The expanded form is \(a^8 b^4 - 6a^4 b^2 + 9\).

Step by step solution

01

- Identify the Binomial

Recognize that the given expression is a binomial raised to the power of 2. It is of the form \(a - b\)^2 where \(a = a^4 b^2\) and \(b = 3\).
02

- Apply the Binomial Square Formula

The formula for the square of a binomial \((a - b)^2\) is \(a^2 - 2ab + b^2\). Substitute \(a = a^4 b^2\) and \(b = 3\) into this formula.
03

- Calculate Each Term

Calculate each term in \((a^4 b^2)^2 - 2(a^4 b^2)(3) + (3)^2\): 1. \(a^2 = (a^4 b^2)^2 = a^8 b^4\) 2. \(-2ab = -2(a^4 b^2)(3) = -6a^4 b^2\) 3. \(b^2 = (3)^2 = 9\)
04

- Combine the Results

Combine the calculated terms to obtain the final expression: \(a^8 b^4 - 6a^4 b^2 + 9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

binomial expressions
A binomial expression is an algebraic expression that contains exactly two distinct terms. These terms are typically separated by a plus or minus sign. In this exercise, the binomial is identified as a^4 b^2 - 3It's helpful to recognize these patterns since they鈥檙e fundamental to solving problems involving binomials. Recognizing a binomial makes it easier to apply specific algebraic formulas, such as the Binomial Square Formula, which is crucial in simplifying expressions raised to a power.
binomial square formula
The Binomial Square Formula is an essential algebraic tool used to square binomial expressions quickly. The formula states: (a - b)^2 = a^2 - 2ab + b^2 This formula allows you to expand expressions without the need to manually multiply the binomial by itself. In this exercise, the binomial a^4 b^2 - 3is squared using this formula. When we apply the formula, a = a^4 b^2 and b = 3. The next step is to calculate each of the terms separately and then combine them.
algebraic multiplication
Algebraic multiplication involves multiplying algebraic terms and expressions, following certain rules. Here, we need to perform algebraic multiplication to expand the binomial square: (a^4 b^2)^2 - 2(a^4 b^2)(3) + (3)^2 Breaking this down:
  • (a^4 b^2)^2 results in a^8 b^4,
  • -2(a^4 b^2)(3) simplifies to -6a^4 b^2,
  • and (3)^2 equals 9.
Combining these results gives us the expanded form of the binomial square: a^8 b^4 - 6a^4 b^2 + 9.

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