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Chapter 5: Problem 34

Combine like terms. $$ 8 r^{5}-2 r^{5} $$

Short Answer

Expert verified
6r^5

Step by step solution

01

Identify Like Terms

Observe the given expression: \(8r^{5} - 2r^{5}\). Both terms are like terms because they have the same variable raised to the same power, which is \(r^5\).
02

Combine the Coefficients

Add or subtract the coefficients (numerical parts) of the like terms. In this case, subtract the coefficient of the second term from the coefficient of the first term: \(8 - 2 = 6\).
03

Rewrite the Expression

Multiply the resulting coefficient by the common variable term: \(6r^{5}\). This is the simplified form.

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Key Concepts

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

Coefficients
In algebra, coefficients are the numerical part of terms that include variables. They are the numbers we multiply by the variables. For example, in the term \(5x\), \(5\) is the coefficient, and \(x\) is the variable. Coefficients can be positive, negative, or zero.

Whenever we combine like terms, we focus on their coefficients first. By adding or subtracting these coefficients, we can simplify the expression. Let's see this with an example:
  • For the expression \(8r^5 - 2r^5\), the coefficients are \(8\) and \(-2\).
    By subtracting \(2\) from \(8\), we get \(6\).
Like Terms
Like terms are terms that have the exact same variables raised to the same powers. They can be combined through addition or subtraction to simplify an expression. For instance, \(3x^2\) and \(7x^2\) are like terms because they both have \(x^2\).

Terms like \(4xy\) and \(10xy\) are also like terms because they share the same variable combination \(xy\). However, \(3x\) and \(4y\) are not like terms as they have different variables.
For the given exercise, both \(8r^5\) and \(2r^5\) are like terms because they each have \(r\) raised to the power of 5. When identifying like terms:
  • Make sure the variables and their exponents are identical.
  • Different variables or exponents mean the terms are not like terms and cannot be combined.
Simplifying Expressions
Simplifying expressions involves combining like terms and performing arithmetic operations to rewrite the expression in its simplest form. Here鈥檚 a step-by-step approach:

1. **Identify like terms:** Look for terms with the same variable and the same power.
2. **Combine the coefficients:** Add or subtract the coefficients of the like terms.
3. **Rewrite the expression:** Use the result to write a simpler form of the original expression.

Let's put these steps into practice using the exercise provided:
  • Given \(8r^5 - 2r^5\), identify like terms which both have \(r^5\).
  • Combine the coefficients: \(8 - 2 = 6\).
  • Rewrite the expression as \(6r^5\).
Simplifying expressions makes algebra easier by reducing complex expressions into a more manageable form. Ensuring each step is done correctly helps in avoiding errors.

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Most popular questions from this chapter

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Write each number in scientific notation. $$ 530 $$
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