Problem 13 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$5 a^{2} b-a b^{2}-7 a^{2} b$$

Step-by-Step Solution

Answer

The simplified expression is \(-2a^2b - ab^2\).

Step 1: Identify Like Terms

First, we examine the expression to identify any like terms. A like term in algebra means terms that have the same variable part raised to the same power. In this given expression, the terms involving both variables are: 1. \( 5a^2b \)2. \( -ab^2 \)3. \( -7a^2b \) The terms \( 5a^2b \) and \( -7a^2b \) are like terms as both have the same variables raised to the same power: \(a^2b\).

Step 2: Combine Like Terms

Now, let's combine the like terms from the identified group:- Combine the coefficients of \(a^2b\): \( 5a^2b - 7a^2b = (5 - 7)a^2b = -2a^2b \)There is no other \(-ab^2\) term to combine, so it remains as is.

Step 3: Write the Simplified Expression

Once we combine the like terms, the simplified expression is:\[ -2a^2b - ab^2 \]This is the simplest form of the expression, with like terms combined.

Key Concepts

Like Terms

In algebra, "like terms" refer to terms that have the exact same variables with the same exponents. Identifying like terms is essential because only like terms can be combined to simplify an expression. Consider the expression given:

• \(5a^2b\)
• \(-ab^2\)
• \(-7a^2b\)

In this expression, both \(5a^2b\) and \(-7a^2b\) are considered like terms because they both include the variables \(a\) and \(b\), where \(a\) is squared \((a^2)\) and \(b\) is raised to the power of 1. The term \(-ab^2\) does not share this variable structure. It has \(b\) squared, meaning it stands alone. Understanding this concept helps simplify equations efficiently and avoid mistakes.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form by performing operations and combining like terms when possible. Let's consider what this means in practical terms. Once we identify like terms, the next step is to simplify:

• For terms that are like, we add or subtract their coefficients.
• Other terms that don't match remain unchanged.

In the case of our example, \(5a^2b\) and \(-7a^2b\) both share the same variable power setup, so we simplify them by combining their coefficients:
\[5 - 7 = -2\]This gives us \(-2a^2b\) as the combined term. Simplification makes problems more manageable and tidier, easing further calculations or analysis.

Combining Like Terms

Combining like terms is a crucial step in simplifying expressions. Once you identify the like terms, you organize them by matching variable parts and processes that involve:

• Adding or subtracting the numeric coefficients of these terms.
• Maintaining the same variable and exponent form.

In the provided exercise, we need to combine the like terms \(5a^2b\) and \(-7a^2b\). This is done by subtracting the coefficients:
\[5 - 7 = -2\]The result \(-2a^2b\) correctly maintains the variable form \(a^2b\). The remaining term in the original expression, \(-ab^2\), does not combine with others and is included in the final simplified expression as is. Combining like terms streamlines solving algebraic expressions, ensuring they're as concise as possible.