Combine Like Terms In 10r²s³ + 42r²s + 19r²s A Step-by-Step Guide

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In the realm of algebra, simplifying expressions is a fundamental skill. It allows us to make complex equations more manageable and easier to understand. One crucial technique in simplifying algebraic expressions is combining like terms. This article delves into the process of identifying and combining like terms, providing a step-by-step guide to simplify the expression 10r²s³ + 42r²s + 19r²s.

Understanding Like Terms

Before we dive into the specific expression, it's essential to grasp the concept of like terms. Like terms are terms that have the same variables raised to the same powers. This means they have the same variable parts, even if their coefficients (the numbers in front of the variables) are different. For example, 3x²y and -5x²y are like terms because they both have the variables x and y raised to the powers of 2 and 1, respectively. However, 2x²y and 4xy² are not like terms because the powers of x and y are different.

To effectively identify like terms, we need to carefully examine the variable parts of each term. Let's break down the expression 10r²s³ + 42r²s + 19r²s to illustrate this:

  • 10r²s³: This term has the variables r and s, raised to the powers of 2 and 3, respectively.
  • 42r²s: This term has the variables r and s, raised to the powers of 2 and 1, respectively.
  • 19r²s: This term has the variables r and s, raised to the powers of 2 and 1, respectively.

Notice that the terms 42r²s and 19r²s have the same variable parts (r²s). This makes them like terms, while 10r²s³ is not a like term with the other two because it has a different power for the variable s.

Step-by-Step Simplification

Now that we've identified the like terms, let's proceed with the simplification process:

  1. Identify Like Terms: As we discussed earlier, the like terms in the expression 10r²s³ + 42r²s + 19r²s are 42r²s and 19r²s.
  2. Combine Coefficients: To combine like terms, we simply add or subtract their coefficients while keeping the variable part the same. In this case, we need to add the coefficients of 42r²s and 19r²s. 42 + 19 = 61. Therefore, the combined term is 61r²s.
  3. Rewrite the Expression: Now, we rewrite the original expression by replacing the like terms with their combined form. The simplified expression becomes 10r²s³ + 61r²s.

The Simplified Expression

After combining the like terms, the simplified expression is 10r²s³ + 61r²s. This expression is more concise and easier to work with than the original expression. We have successfully reduced the number of terms by combining those that share the same variable parts.

Importance of Combining Like Terms

Combining like terms is a fundamental skill in algebra for several reasons:

  • Simplifying Expressions: It makes complex expressions more manageable and easier to understand. This is crucial for solving equations, graphing functions, and performing other algebraic operations.
  • Reducing Errors: By simplifying expressions, we reduce the chances of making errors in subsequent calculations. A simpler expression has fewer terms and operations, minimizing the potential for mistakes.
  • Improving Efficiency: Simplified expressions are easier to work with and can save time in problem-solving. They allow us to focus on the essential parts of the problem without being bogged down by unnecessary complexity.
  • Facilitating Further Operations: Simplifying expressions often makes it easier to perform other algebraic operations, such as factoring, expanding, and evaluating expressions.

Additional Examples

To further solidify your understanding of combining like terms, let's look at a couple more examples:

Example 1: Simplify the expression 5x² + 3x - 2x² + 7x - 4.

  1. Identify Like Terms: The like terms are 5x² and -2x², as well as 3x and 7x.
  2. Combine Coefficients:
    • 5x² - 2x² = 3x²
    • 3x + 7x = 10x
  3. Rewrite the Expression: The simplified expression is 3x² + 10x - 4.

Example 2: Simplify the expression 8ab - 4a²b + 2ab² + 6a²b - 3ab.

  1. Identify Like Terms: The like terms are 8ab and -3ab, as well as -4a²b and 6a²b.
  2. Combine Coefficients:
    • 8ab - 3ab = 5ab
    • -4a²b + 6a²b = 2a²b
  3. Rewrite the Expression: The simplified expression is 5ab + 2a²b + 2ab².

Common Mistakes to Avoid

While combining like terms is a straightforward process, there are some common mistakes that students often make. Being aware of these mistakes can help you avoid them:

  • Combining Unlike Terms: The most common mistake is combining terms that are not like terms. Remember, terms must have the same variables raised to the same powers to be considered like terms. For example, you cannot combine 3x² and 2x because they have different powers of x.
  • Forgetting the Sign: When combining coefficients, it's crucial to pay attention to the signs (positive or negative) of the terms. For example, if you have -5x² + 2x², the result is -3x², not 7x².
  • Incorrectly Adding/Subtracting Coefficients: Make sure you are adding or subtracting the coefficients correctly. Double-check your calculations to avoid errors.
  • Changing the Variable Part: When combining like terms, the variable part remains the same. Only the coefficients are added or subtracted. For example, 4x² + 3x² = 7x², not 7x⁴.

Practice Problems

To master the skill of combining like terms, it's essential to practice. Here are some practice problems for you to try:

  1. Simplify: 7y³ - 2y² + 5y³ + y² - 3y
  2. Simplify: 9p²q - 4pq² + 6p²q + 2pq² - pq
  3. Simplify: 12m⁴n - 5m²n² + 3m⁴n - 8m²n² + 2mn⁴

Try to solve these problems on your own, and then check your answers with the solutions provided below:

Solutions:

  1. 12y³ - y² - 3y
  2. 15p²q - 2pq² - pq
  3. 15m⁴n - 13m²n² + 2mn⁴

Conclusion

Combining like terms is a fundamental skill in algebra that simplifies expressions and makes them easier to work with. By understanding the concept of like terms and following the step-by-step process outlined in this article, you can confidently simplify algebraic expressions and avoid common mistakes. Remember to practice regularly to solidify your skills and improve your efficiency. The ability to combine like terms is a crucial building block for more advanced algebraic concepts, so mastering it will set you up for success in your mathematical journey.

By simplifying the expression 10r²s³ + 42r²s + 19r²s, we arrived at the simplified form of 10r²s³ + 61r²s. This process demonstrates the power of combining like terms to make algebraic expressions more manageable and understandable. Keep practicing, and you'll become a pro at simplifying expressions in no time!

Combine like terms in the expression: 10r²s³ + 42r²s + 19r²s. If there are no like terms after simplification, rewrite the simplified expression.

Combining Like Terms 10r²s³ + 42r²s + 19r²s A Step-by-Step Guide