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Chapter 1: Problem 30

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

Short Answer

Expert verified
Unlike terms.

Step by step solution

01

Understand the Definition

Like terms are terms that have the same variable raised to the same power. Unlike terms have different variables or the same variables raised to different powers.
02

Identify the Variables and Exponents

Look at the variables and their exponents in each term. For the terms given: 1. For the first term: \(8x^{5}\), the variable is \(x\) and the exponent is 5. 2. For the second term: \(-10x^{3}\), the variable is \(x\) and the exponent is 3.
03

Compare the Terms

Compare the exponents of the variables in both terms. The first term has an exponent of 5, and the second term has an exponent of 3.
04

Determine if the Terms are Like or Unlike

Since the exponents are different (5 in the first term and 3 in the second term), the terms are unlike.

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Key Concepts

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

Variables
In algebra, a variable is a symbol, usually a letter, that represents a number. Variables are important because they allow us to write expressions and equations in a general form. For example, in the term \(8x^{5}\), the variable is the letter 'x'.
Variables make it possible to solve problems with unknown values and to develop general formulas. When comparing terms, identifying the variables is crucial.
If the variables are different, then the terms are automatically unlike, regardless of any other factors.
Exponents
Exponents indicate how many times a number (called the base) is multiplied by itself. In the term \(8x^5\), the 5 is the exponent and it means that 'x' is multiplied by itself 5 times (\(x\)\times\(x\)\times\(x\)\times\(x\)\times \(x\)).
Exponents are very important in determining whether terms are like or unlike. For terms to be like, they must not only have the same variables but also the same exponents.
In our exercise, the first term with \(5\) as the exponent and the second term with \(3\) as the exponent differ, making them unlike.
Terms Comparison
To determine if terms are like or unlike, we must compare both the variables and the exponents.
  • First, check the variables in each term. If the variables are different, the terms are unlike.
  • Second, if the variables are the same, compare the exponents. Different exponents mean the terms are unlike.
  • If both the variables and exponents match, then the terms are like.
In our exercise, both terms (\(8x^5\)\times \(-10x^3\)) have the same variable 'x', but their exponents are different (5 and 3). Therefore, they are unlike terms.
Remember, identifying whether terms are like or unlike helps in simplifying expressions and solving equations efficiently.

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