Question

Simplify by combining like terms. $$-14 u^{3}+13 u^{3}$$

Short answer

Question: Simplify the expression: (-14u^3) + (13u^3) Answer: -u^3

Step-by-step solution

Identify the like terms.

In the given expression, both terms are like terms because they both contain \(u^3\) variables: \((-14u^3)\) and \((13u^3)\).

Combine the like terms.

To combine the like terms, we will add their coefficients. In this case, we need to add \((-14)\) and \((13)\). So, \((-14) + (13) = -1\).

Write the simplified expression.

Now that we have combined the like terms, we can write the simplified expression: $$-1u^3$$ or simply, $$-u^3$$.

Key concepts

Simplify Expressions

When we talk about simplifying expressions in algebra, we refer to the process of making an algebraic expression as straightforward as possible. This means reducing the complexity of the expression by combining like terms, eliminating unnecessary parentheses, and performing any arithmetic operations. The goal is to end up with an expression that is easier to understand and work with.

For example, consider the expression \( -14u^3 + 13u^3 \). The simplification process involves identifying like terms, which are terms that can be combined because they have the same variables raised to the same powers. In our case, both terms have \(u^3\) as the variable part. The next step is to merge these like terms by adding their coefficients, which leads to the simplified expression \( -u^3 \). Simplifying expressions is a foundational skill in algebra that allows students to solve equations more efficiently and understand how different terms relate to each other.

Like Terms in Algebra

Understanding like terms in algebra is crucial for simplifying expressions correctly. Like terms are terms within an algebraic expression that have identical variable parts when the variables are raised to the same power. It's important to note that the numerical coefficients can be different, but the variable components must match for the terms to be considered 'like' terms.

Consider our textbook exercise's terms, \( -14u^3 \) and \( 13u^3 \). Despite having different coefficients, because the variable part \(u^3\) is the same for both, they are indeed like terms. We simplify the expression by combining these terms. To do this, we add or subtract the coefficients while keeping the variable part unchanged. This process simplifies the expression to a single term, \( -u^3 \) in our case, making it much more manageable to work with in further mathematical operations.

Coefficients in Algebra

In the realm of algebra, a coefficient is a numerical factor that multiplies a variable in an algebraic expression. It is an essential part of any term and helps determine the term's value. In the expression \( -14u^3 + 13u^3 \), the numbers \( -14 \) and \( 13 \) are the coefficients of the terms \(u^3\).

When combining like terms, as we did in our exercise, we focus solely on the coefficients. We treat the variable part as a common factor and simply perform arithmetic operations on the coefficients. In our example, combining the coefficients \( -14 \) and \( 13 \) gives us the result \( -1 \) when added together. This result is then used as the new coefficient of the common variable part, \(u^3\), leading to our final simplified expression \( -u^3 \). Understanding coefficients is vital for navigating through algebraic expressions and enables students to tackle more complex algebraic manipulations.