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Chapter 5: Problem 120

Are \(5 x^{2}\) and \(5 x^{3}\) like terms? Explain.

Short Answer

Expert verified
No, 5x虏 and 5x鲁 are not like terms because their exponents are different.

Step by step solution

01

Understand Like Terms

Like terms are terms that have the exact same variable part, including both the variable itself and its exponent. Only the coefficients (numbers in front) can be different.
02

Identify the Terms

The given terms are 5x虏 and 5x鲁. Observe the variable part of each term.
03

Compare Variable Parts

Compare the exponents of the variable x in each term. In 5x虏, the exponent is 2. In 5x鲁, the exponent is 3.
04

Determine if They are Like Terms

Since the exponents of the variable x are different (2 in 5x虏 and 3 in 5x鲁), the terms are not like terms.

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Key Concepts

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

Variable Parts
When you look at algebraic expressions, you'll often come across the term 'variable parts.' But what exactly are they?

Variable parts include both the variables, like x, y, or z, and their associated exponents.
For example, in 5x虏, 'x虏' is the variable part. It's important to note this because identifying like terms depends on these variable parts.

Like terms have the same variable parts, meaning they have the same variable with the same exponent. If either the variable or the exponent differs, the terms are not like terms.

Identifying variable parts can help you to:
  • Simplify expressions
  • Combine like terms
  • Solve algebraic equations
Exponents
Exponents are the small numbers written slightly above and to the right of a variable. They indicate how many times the variable is multiplied by itself.

For instance, in the term 5x虏, the exponent is 2. This means x is multiplied by itself: x * x.

Understanding exponents is crucial for distinguishing like terms because the exponents must match for the terms to be considered 'like.'

Key points about exponents:
  • An exponent of 2 means 'squared'
  • An exponent of 3 means 'cubed'
  • An exponent of 1 is usually not shown but is implied

Let's look at an example:
  • In 5x虏, the exponent is 2
  • In 5x鲁, the exponent is 3
These terms are not like terms because the exponents are different.
Coefficients
The coefficient is the number that is in front of the variable part in an algebraic expression.
In 5x虏, the coefficient is 5. It's basically a multiplier for the variable part.

Unlike variable parts and exponents, coefficients can be different and still have terms be considered 'like.'

Let's explore coefficients a bit more:
  • In the term 7y鲁, 7 is the coefficient
  • In the term -2x, -2 is the coefficient
  • The coefficient tells you how many of the variable parts you have

It's useful to remember that while coefficients can differ, the variable part and the exponent must be the same for terms to be like terms.

For example, 5x虏 and 3x虏 are like terms because they have the same variable part (x虏) but different coefficients (5 and 3).

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