Problem 73 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ 3 r^{4}-4 r+7 r^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(10r^4 - 4r\).

Step 1: Identify like terms

In the expression, the terms are: \(3r^4\), \(-4r\), and \(7r^4\). The like terms are \(3r^4\) and \(7r^4\) because they both have the same variable part \(r^4\).

Step 2: Combine like terms

Add the coefficients of the like terms: \(3r^4 + 7r^4 = (3+7)r^4 = 10r^4\).

Step 3: Write the simplified expression

After combining the like terms, the simplified expression is \(10r^4 - 4r\).

Key Concepts

Like Terms

In algebra, understanding 'like terms' is key to simplifying expressions effectively. Each 'like term' in a polynomial expression shares the same variables raised to the same powers. Essentially, these terms have identical variable parts. When examining the expression \(3r^4 - 4r + 7r^4\), you'll see different components.

• The term \(3r^4\) has the variable \(r\) raised to the fourth power.
• The term \(-4r\) has the variable \(r\) raised to the first power.
• The term \(7r^4\) also has the variable \(r\) raised to the fourth power.

The like terms here are \(3r^4\) and \(7r^4\), both having \(r^4\). Recognizing these helps in the next steps of combining them.

Combining Like Terms

Once you have identified like terms, the next step is to combine them. Combining like terms involves adding or subtracting their coefficients while keeping the variable part the same.

For example, if you have \(3r^4 + 7r^4\), you only add the numbers in front of \(r^4\):

• Coefficient for \(3r^4\) is 3.
• Coefficient for \(7r^4\) is 7.

Adding these gives you \(3 + 7 = 10\), so \(3r^4 + 7r^4\) simplifies to \(10r^4\).

The key is keeping the variable part the same and changing only the coefficients. This method helps simplify expressions for easier calculations or further operations.

Simplifying Expressions

With like terms combined, the final task is simplifying expressions. This streamlines an expression, making it more efficient and manageable. In our exercise, after combining like terms, the expression simplified to \(10r^4 - 4r\).

The expression initially had three terms, but realizing that \(3r^4\) and \(7r^4\) could be combined into \(10r^4\) reduced its complexity. Now, it has only two terms:

• \(10r^4\) - a term with \(r\) to the fourth power.
• \(-4r\) - a linear term with \(r\).

Simplifying makes expressions more concise and often helps in solving equations or understanding their behavior more quickly. Always strive to combine and reduce until no further simplification is possible.