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Chapter 2: Problem 103

Discuss whether \(7 x x y^{3}\) and \(5 x^{2} y y y\) are like terms.

Short Answer

Expert verified
Yes, \(7 x x y^{3}\) and \(5 x^{2} y y y\) are like terms.

Step by step solution

01

Identify Variables and Powers in Term 1

Here in the term \(7 x x y^{3}\), there are two variables, x and y. The power of x is 2 (since there are two x's multiplied together) and the power of y is 3.
02

Identify Variables and Powers in Term 2

In the second term \(5 x^{2} y y y\), again there are two variables x and y. The power of x is 2 and the power of y is also 3 (because there are three y's multiplied together).
03

Compare Variables and Powers

Comparing the variables and their powers in both the terms, we see that for the variable x, the power is 2 in both the terms. Similarly, for the variable y, the power is 3 in both the terms.
04

Conclusion

Since in both terms, the variables and their respective powers are the same (2 for x and 3 for y), the terms \(7 x x y^{3}\) and \(5 x^{2} y y y\) are like terms.

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Key Concepts

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

Understanding Variables
Variables are symbols used in algebra to represent unknown quantities. The most common symbols are letters like x, y, or z. In the terms from the exercise, x and y represent variables. Think of variables as placeholders for numbers that we may not know yet. This allows us to form expressions and equations that can be manipulated and solved.
  • Variables provide flexibility in mathematical expressions.
  • They allow the representation of general patterns and relationships.
Identifying variables correctly is crucial in deciding if terms can be grouped together, especially in polynomials and algebraic expressions.
The Role of Powers in Algebra
Powers, also known as exponents, are used to indicate repeated multiplication of a variable by itself. For example, in the expression "x squared" or \(x^2\), x is multiplied by itself once (x times x). Similarly, in the expression \(y^3\), y is used three times in multiplication (y times y times y).
  • Powers determine the degree of each variable in a term.
  • Powers must match for terms to be considered 'like terms'.
Understanding powers is a key step in comparison processes. When two algebraic terms are being evaluated as "like terms", their variables must not only match but also their respective powers must be the same.
Algebraic Expressions and Like Terms
Algebraic expressions are combinations of variables, numbers, and mathematical operations (like addition or multiplication). Each combination forms a 'term', such as \(7x^2y^3\) or \(5x^2y^3\).
To determine if terms are 'like terms':
  • The variables and their powers must match.
  • The coefficients (numerical parts) do not need to be the same.
In the exercise, both terms \(7xx y^3\) and \(5x^2 yyy\) can be simplified further to reveal \(7x^2y^3\) and \(5x^2y^3\), showing that the variables and their exponents align perfectly. Hence, these terms are like terms, allowing for simplification in algebraic expressions.

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