Problem 20 - Introductory Algebra for College Students

Question

In Exercises \(15-28,\) simplify each algebraic expression, or explain why the expression cannot be simplified. $$14 x^{4}+x^{4}$$

Step-by-Step Solution

Answer

The simplification of the expression \(14 x^{4}+x^{4}\) is \(15 x^{4}\).

Step 1: Identifying the like terms

In the expression \(14 x^{4}+x^{4}\), the terms \(14 x^{4}\) and \(x^{4}\) are like terms because they both contain the variable \(x^{4}\) raised to the same power.

Step 2: Simplifying the expression

The process of simplification involves combining the coefficients of the like terms. Here, the coefficient of \(x^{4}\) is 1. When we add this to the coefficient of \(14 x^{4}\), we get \(14 + 1 = 15\). Therefore, the simplification of \(14 x^{4}+x^{4}\) gives us \(15 x^{4}\).

Key Concepts

Like Terms in Algebra

Algebra is full of symbols, but understanding the concept of like terms is key to simplifying expressions. Like terms are terms within an algebraic expression that have the same variables raised to the same powers. Think of them like identical twins – they look the same and can easily be combined. For instance, in the expression \(14 x^{4}+x^{4}\), both terms involve the variable \(x\) raised to the fourth power. Since they are