Problem 64 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-9 m^{3}+3 m^{3}-7 m^{3}\)

Step-by-Step Solution

Answer

-13m^{3}

Step 1: Identify like terms

Notice that all the terms \( -9m^{3}, 3m^{3}, \text{ and } -7m^{3} \) have the same variable part \( m^{3} \). Therefore, they are like terms and can be combined.

Step 2: Combine like terms

Add the coefficients of the like terms: \( -9 + 3 - 7 \). Simplify the addition: \( -9 + 3 \) equals \( -6 \), and \( -6 - 7 \) equals \( -13 \).

Step 3: Write the simplified expression

Combine the result with the variable part: \( -13m^{3} \). Thus, the simplified expression is \( -13m^{3} \).

Key Concepts

Like Terms in Algebra

In algebra, like terms are terms that have the same variable raised to the same power. For instance, in the exercise we have \( -9m^{3}, 3m^{3}, \text{ and } -7m^{3} \). All these terms have the variable \( m^{3} \), making them like terms. Recognizing like terms is crucial as it enables you to simplify expressions easily. Always look for terms that share both the same variable and exponent.

Combining Like Terms

Once you've identified like terms, the next step is to combine them. Combining like terms involves adding or subtracting their coefficients. For example, in the exercise, we have the coefficients -9, 3, and -7 for the like terms \( m^{3} \).

• Start by arranging the coefficients to be combined: -9 + 3 - 7.
• First, add -9 and 3 to get -6.
• Then, subtract 7 from -6 to get -13.

By combining the coefficients this way, we get the simplified coefficient for our \( m^{3} \) term.

Polynomial Simplification

Simplifying polynomials involves combining like terms to write the expression in its simplest form. After combining like terms, we attach the simplified coefficient to the common variable. In our exercise, the simplified coefficient was -13. Therefore, attaching this to \( m^{3} \) gives us the final expression: \( -13m^{3} \).

• Identify like terms.
• Combine their coefficients.
• Attach the result to the common variable.

Following these steps will always help you to simplify polynomial expressions accurately.