Problem 53 - Prealgebra

Question

Combine like terms. $$2 a b+5 a b$$

Step-by-Step Solution

Answer

The simplified expression is \(7ab\).

Step 1: Identify Like Terms

Look at the expression given: \(2ab + 5ab\). Both terms contain the same variables 'a' and 'b' raised to the same powers, so they are like terms.

Step 2: Combine Like Terms

Since both terms are like terms, we can add their coefficients together. The coefficients are the numbers in front of the variables: 2 and 5. Adding these gives us: \(2 + 5 = 7\).

Step 3: Write the Simplified Expression

Now that we know the sum of the coefficients is 7, we write the simplified expression as \(7ab\).

Key Concepts

Coefficients

Let's start by demystifying what a coefficient is. In algebraic expressions, a coefficient is the number that is multiplied by a variable or a product of variables. Consider the term \(2ab\). Here, the number \(2\) is the coefficient.
It indicates that \(ab\) is being multiplied by \(2\). When we deal with like terms, the coefficients help us in determining how many sets of a particular variable-product we have.
For example, in the expression \(2ab + 5ab\), \(2\) and \(5\) are the coefficients, telling us the expression represents \(2\) lots of \(ab\) and \(5\) more lots of \(ab\).
When combining like terms, it's crucial to pay attention to these coefficients, as they determine the contribution each term makes to the total.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves combining like terms, substituting in values, and using algebraic rules to make an expression easier to work with.
In our exercise, when given an expression like \(2ab + 5ab\), the goal of simplifying is to combine terms wherever possible. This process often makes it easier to understand, evaluate, or further manipulate the expression.

• Identify any like terms that can be combined, like we do with \(2ab\) and \(5ab\).
• Add or subtract the coefficients of these like terms.
• Rewrite the expression with the new simplified term, \(7ab\), in this case.

By simplifying expressions, we eliminate unnecessary complexity, making it much easier for future computation and problem-solving.

Like Terms in Algebra

An essential concept in algebra is recognizing like terms. Like terms are components of an expression that have the same variable parts. This means they have the same variable(s) raised to the same powers.
Take \(2ab\) and \(5ab\). They are like terms because both include the variables \(a\) and \(b\) to the same power. This is why we can combine them.
Here’s how you can identify and handle like terms:

• Inspect the variables and their exponents in each term.
• If variables and exponents match, you've got like terms.
• Add or subtract their coefficients, as needed.

By combining like terms, we make expressions more concise and easier to handle. This is especially helpful in solving equations and simplifying expressions.