Comparison of Variables and Exponents

Oct 13, 2025 by windailys.com 23 views

What are Like Terms?

In algebra, like terms are expressions that have the same variables raised to the same powers. This means that the variables and their exponents must be identical in both expressions. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

The Case of Lena's Statement

Lena says that $$4xy^3$$ and $$-5x^3y$$ are like terms. To determine if she is correct, we need to examine the variables and their exponents in both expressions.

Variables and Exponents

In the expression $$4xy^3$$, the variable $x$ is raised to the power of 1, and the variable $y$ is raised to the power of 3. In the expression $$-5x^3y$$, the variable $x$ is raised to the power of 3, and the variable $y$ is raised to the power of 1.

Comparison of Variables and Exponents

As we can see, the variables and their exponents in the two expressions are not identical. The variable $x$ is raised to different powers in each expression, and the variable $y$ is also raised to different powers. Therefore, the expressions $$4xy^3$$ and $$-5x^3y$$ are not like terms.

Conclusion

Based on the definition of like terms, we can conclude that Lena is incorrect. The expressions $$4xy^3$$ and $$-5x^3y$$ are not like terms because the same variables are not raised to the same powers.

Answer

The correct answer is A. No, because the same variables are not raised to the same powers.

Why is this Important?

Understanding like terms is crucial in algebra because it allows us to combine expressions using the rules of addition and subtraction. When we have like terms, we can add or subtract their coefficients (the numbers in front of the variables) to simplify the expression. However, when we have unlike terms, we cannot combine them in this way.

Example

Consider the expression $$2x^2 + 3x^2$$. In this case, we have like terms because both expressions have the variable $x$ raised to the power of 2. We can combine the coefficients to get $$5x^2$$.

On the other hand, consider the expression $$2x^2 + 3y^2$$. In this case, we have unlike terms because the variables $x$ and $y$ are raised to different powers. We cannot combine the coefficients in this case.

Real-World Applications

Understanding like terms has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, we often need to combine forces or energies using the rules of addition and subtraction. In engineering, we may need to combine materials or stresses using the rules of addition and subtraction. In economics, we may need to combine costs or revenues using the rules of addition and subtraction.

Conclusion

Q: What are like terms in algebra?

A: Like terms are expressions that have the same variables raised to the same powers. This means that the variables and their exponents must be identical in both expressions.

Q: How do I determine if two expressions are like terms?

A: To determine if two expressions are like terms, you need to examine the variables and their exponents in both expressions. If the variables and their exponents are identical, then the expressions are like terms.

Q: What is the difference between like terms and unlike terms?

A: Like terms are expressions that have the same variables raised to the same powers, while unlike terms are expressions that have different variables or different exponents.

Q: Can I add or subtract like terms?

A: Yes, you can add or subtract like terms. When you have like terms, you can combine their coefficients (the numbers in front of the variables) to simplify the expression.

Q: Can I add or subtract unlike terms?

A: No, you cannot add or subtract unlike terms. Unlike terms cannot be combined using the rules of addition and subtraction.

Q: What is an example of like terms?

A: An example of like terms is $$2x^2 + 3x^2$$. In this case, we have like terms because both expressions have the variable $x$ raised to the power of 2.

Q: What is an example of unlike terms?

A: An example of unlike terms is $$2x^2 + 3y^2$$. In this case, we have unlike terms because the variables $x$ and $y$ are raised to different powers.

Q: Why is it important to understand like terms?

A: Understanding like terms is crucial in algebra because it allows us to combine expressions using the rules of addition and subtraction. When we have like terms, we can add or subtract their coefficients to simplify the expression.

Q: How do I apply like terms in real-world situations?

A: Like terms have many real-world applications in fields such as physics, engineering, and economics. For example, in physics, we often need to combine forces or energies using the rules of addition and subtraction. In engineering, we may need to combine materials or stresses using the rules of addition and subtraction. In economics, we may need to combine costs or revenues using the rules of addition and subtraction.

Q: Can I use like terms to simplify complex expressions?

A: Yes, you can use like terms to simplify complex expressions. By combining like terms, you can simplify the expression and make it easier to work with.

Q: What are some common mistakes to avoid when working with like terms?

A: Some common mistakes to avoid when working with like terms include:

Not identifying like terms correctly
Not combining like terms correctly
Not simplifying expressions correctly

Q: How can I practice working with like terms?

A: You can practice working with like terms by:

Solving problems that involve like terms
Combining like terms in different expressions
Simplifying complex expressions using like terms

Conclusion

In conclusion, like terms are expressions that have the same variables raised to the same powers. Understanding like terms is crucial in algebra because it allows us to combine expressions using the rules of addition and subtraction. By practicing working with like terms, you can improve your skills and apply them to real-world situations.