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Chapter 2: Problem 97

In the following exercises, identify all sets of like terms. $$x^{3}, 8 x, 14,8 y, 5,8 x^{3}$$

Short Answer

Expert verified
The sets of like terms are: \(x^3\) and \(8x^3\); \(8x\); \(14\) and \(5\); \(8y\).

Step by step solution

01

Identify Like Terms

Like terms are terms that contain the same variables raised to the same power. Examine each pair of terms to see if they match in this respect.
02

Compare the Terms

Compare each term to identify sets of like terms:- Terms with variable and power: - For terms with variable and exponent, identify the variable part: 1. \(x^3\) 2. \(8x\) 3. \(8y\) 4. \(8x^3\)- Constants: 1. \(14\) 2. \(5\)
03

Group Like Terms

Group the identified like terms:- Like terms with \(x^3\): 1. \(x^3\) 2. \(8x^3\)- Like terms with \(x\): 1. \(8x\)- Constants: 1. \(14\) 2. \(5\)- Like terms with \(y\): 1. \(8y\)

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Key Concepts

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

identifying like terms
In algebra, identifying like terms is crucial. Like terms are those that have identical variables raised to the same exponent. This means the variable and its exponent must be exactly the same for the terms to be considered 'like'.

For example, in the list of terms given: \(x^{3}, 8x, 14, 8y, 5, 8x^{3}\), you can spot that terms such as \(x^{3}\) and \(8x^{3}\) are like terms because they share the same variable, \(x\), raised to the same power, 3.

Remember, constants (numbers without variables) are always like terms with each other, because they don't contain any variables in the first place. Thus, in our example, \(14\) and \(5\) are like terms.
variables and exponents
When dealing with terms in algebra, it's important to focus on both the variables and their exponents. The variable is the letter (such as \(x\) or \(y\)), while the exponent is the small number that tells you how many times the variable is multiplied by itself.

To identify like terms, check:
  • The variable: Is it the same letter?
  • The exponent: Is it the same number?
If both are identical, they are like terms. For instance, \(8x\) and \(8y\) are not like terms because their variables are different, even though the coefficients (numbers in front of variables) are the same.
grouping like terms
Once you've identified like terms in a list, the next step is to group them. This helps in simplifying your algebraic expressions.

From our example \(x^{3}, 8x, 14, 8y, 5, 8x^{3}\), we can create groups:
  • Like terms with \(x^3\): \(x^{3}\), \(8x^{3}\)
  • Like terms with \(x\): \(8x\) (there's only one)
  • Constants: \(14\), \(5\)
  • Like terms with \(y\): \(8y\) (there's only one)
Grouping like terms simplifies your calculations and makes it easier to solve algebraic equations.

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