Question

In each case, simplify the given expression, if possible. \(a b-b a\)

Short answer

Question: Simplify the expression \(ab - ba\). Answer: \(0\)

Step-by-step solution

Identify like terms

In the expression \(ab - ba\) we can notice that \(ab = ba\) due to the commutative property of multiplication. This means that we have like terms.

Simplify the expression by applying the commutative property

Since \(ab = ba\), the expression can be simplified to: \(ab - ba = ab - ab\)

Cancel out the terms

Now that we have the expression \(ab - ab\), we can cancel out the terms: \(ab - ab = 0\) The simplified expression is \(0\).

Key concepts

Understanding Like Terms

In algebra, the concept of "like terms" is vital for simplifying expressions. Like terms are terms that have the same variables raised to the same powers. This means the coefficients might differ, but the variable parts must be identical for terms to be considered like. For instance, in the expression \(3x + 5x\), both \(3x\) and \(5x\) are like terms because they both contain the variable \(x\) raised to the same power.
When dealing with expressions, recognizing like terms allows you to combine them. Combining like terms helps in reducing the complexity of an expression, making it simpler and more manageable. In our specific exercise, the terms \(ab\) and \(ba\) were shown to be the same due to the commutative property of multiplication (more on this later). This realization transforms them into like terms, which is the first step in simplifying the expression.

Simplifying Expressions with Basic Properties

Simplifying expressions is all about reducing them to their most basic form. It's like tidying up a cluttered room! In algebra, we use certain properties and rules to achieve this. Among these is the **commutative property of multiplication**, which states that the order in which two numbers are multiplied doesn't affect the product.
For example, \(ab = ba\). This means we can rearrange the multiplication of terms without changing their value. Simplifying involves rewriting expressions to reveal like terms and making cancellations where possible. In the example provided, realizing \(ab\) and \(ba\) are the same allowed us to simplify the expression \(ab - ba\) to \(ab - ab\). This step makes complex equations easier to solve, paving the way for further simplification and cancellation.

• Apply the commutative property to identify like terms.
• Substitute expressions like \(ab - ba\) with \(ab - ab\) when terms are rearranged through properties.

Cancellation in Algebra: A Clearer View

Cancellation is a powerful tool in algebra used to simplify expressions further. Once like terms in an expression have been combined or aligned, any identical terms with opposite signs can cancel each other out. Think of it as finding a zero pair in the equation.
In the expression \(ab - ab\), cancellation is straightforward. Since they are the same term with opposite operations (addition and subtraction), they perfectly cancel each other to result in zero—thus simplifying the entire expression. This process helps in solving equations and finding the simplest form of expressions.

• Cancelling is like a balancing act—opposite terms neutralize each other.
• Understanding this concept makes it easier to tackle larger algebraic expressions.

By recognizing the opportunity for cancellation, you reduce unnecessary complexity and move closer to solving the equation effectively.