Problem 38 - Beginning and Intermediate Algebra

Question

Identify each group of terms as like or unlike. \(8 x^{5},-10 x^{3}\)

Step-by-Step Solution

Answer

The terms \(8x^5\) and \(-10x^3\) are unlike terms.

Step 1 - Identify the Variables and Exponents

Both terms must have the same variable(s) raised to the same power to be considered like terms. In this exercise, the terms are: \(8x^5\) and \(-10x^3\).

Step 2 - Compare the Variables

Examine the variable in each term. Both terms have the variable \(x\). This satisfies the first condition for like terms.

Step 3 - Compare the Exponents

Check the exponents of the variable in each term. The term \(8x^5\) has the exponent 5, while the term \(-10x^3\) has the exponent 3.

Step 4 - Determine if the Terms are Like or Unlike

Since the exponents in \(8x^5\) and \(-10x^3\) are different, the terms do not meet the condition for like terms. Therefore, they are unlike terms.

Key Concepts

Variables

Variables are fundamental in algebra. A variable is a symbol, usually a letter, that represents a number. Its value can change or vary. Think of it as a placeholder for something that can shift.
For example, in the term \(8x^5\), the letter \(x\) is the variable. This means \(x\) can be any number. Variables help us generalize mathematical problems. They allow us to write expressions and equations that describe how quantities are related.
Trouble starts if you confuse different variables. For instance, \(x\) and \(y\) are different variables, and they represent different things. Make sure to always recognize which variable you are dealing with. Identifying the variable correctly is the first step in understanding like and unlike terms.

Exponents

Exponents show how many times a number, called the base, is multiplied by itself. In the term \(8x^5\), the exponent is 5. It means \(x\) is multiplied by itself five times.
Here is how it works: \[x^5 = x \times x \times x \times x \times x \]
The exponent tells us the power of the variable. If you have two terms with the same variable but different exponents, they are not like terms. For example, \(x^5\) and \(x^3\) have the same variable but different exponents. Hence, they are unlike terms. It's crucial to compare exponents carefully to determine if terms are like or unlike.

Algebraic Terms

Algebraic terms are the building blocks of algebra. They can consist of numbers, variables, exponents, and their combinations. Each term is a single part of an expression or equation.
Consider the term \(8x^5\). It includes:

• The coefficient: 8, which is the numerical part.
• The variable: \(x\), representing an unknown value.
• The exponent: 5, showing the power of the variable.

To identify like and unlike terms, algebraic terms must be compared based on their variables and exponents. Terms with the same variables raised to the same powers are like terms.
For instance, \(8x^5\) and \(3x^5\) are like terms. They have the same variable \(x\) and the same exponent 5. However, \(8x^5\) and \(-10x^3\) are unlike terms because they have different exponents. Recognizing these distinctions is key to mastering algebra.