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Chapter 1: Problem 72

Simplify by combining like terms. $$ 32 a-a+5 a $$

Short Answer

Expert verified
The simplified expression is \(36a\).

Step by step solution

01

Identify Like Terms

The expression given is \(32a - a + 5a\). Like terms are terms that contain the same variable raised to the same power. In this expression, all terms \(32a\), \(-a\), and \(5a\) are like terms because they all contain the variable \(a\) to the first power.
02

Combine Coefficients

To combine like terms, add or subtract their coefficients. The expression \(32a - a + 5a\) can be simplified by combining the coefficients of \(a\). This means you calculate \(32 - 1 + 5\).
03

Simplify the Expression

Calculate the sum of the coefficients: \(32 - 1 + 5 = 36\). Thus, the simplified expression of \(32a - a + 5a\) is \(36a\).

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Key Concepts

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

Understanding Like Terms
In algebra, understanding like terms is crucial for simplifying expressions. Like terms are terms that have the same variables raised to the same powers. For instance, in the expression \(32a - a + 5a\), all terms are like terms as they each consist of the variable \(a\) raised to the power of one. Recognizing like terms allows you to combine them more easily. This is important because only like terms can be added or subtracted directly.To spot like terms, follow these tips:
  • Look for terms with identical variables and exponents.
  • Constant terms (those without variables) are also like terms with each other.
By mastering the identification of like terms, you create a strong foundation for more complex algebraic manipulations.
Combining Coefficients
Once you've identified the like terms, the next step is combining their coefficients. The coefficient is the numerical part of a term, which is multiplied by the variable. In our example, the expression \(32a - a + 5a\) involves three terms with \(a\) as the variable. The coefficients here are 32, -1, and 5.Combining coefficients involves simple arithmetic:
  • Simply add or subtract the coefficients of the like terms.
For \(32a - a + 5a\), the operation is \(32 - 1 + 5\). Calculating this gives 36. Thus, the terms combine to form \(36a\).Getting comfortable with combining coefficients makes simplifying algebraic expressions less intimidating.
Simplification of Expressions
Simplification is the process of rewriting an expression in its most efficient and comprehensible form without changing its value. When it comes to expressions like \(32a - a + 5a\), simplification is achieved by combining like terms and reducing the expression to its simplest form, which in this case is \(36a\).Why simplify?
  • Simplifying makes expressions easier to work with, especially in more extensive calculations.
  • It helps in better understanding the underlying mathematics of a problem or equation.
Remember, simplifying doesn't change the value of the expression, it just makes it more straightforward. Through practice, you'll find simplification becomes second nature, leading to more efficient problem solving.

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