Problem 17 - Introductory Algebra for College Students

Question

In Exercises \(15-28,\) simplify each algebraic expression, or explain why the expression cannot be simplified. $$10 x^{3}+5 x^{3}$$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression \(10 x^{3}+5 x^{3}\) is \(15x^{3}\).

Step 1: Identify like terms

In the expression \(10 x^{3}+5 x^{3}\), notice that both terms are 'like terms', meaning they have the same variable (x) raised to the same power (3).

Step 2: Combine like terms

Once you've identified these as like terms, the next step involves combining them, essentially by adding their coefficients. The coefficient in a term refers to the numerical part of the term. Here, the coefficients are 10 and 5 for \(10 x^{3}\) and \(5 x^{3}\) respectively. To add these, simply calculate \(10 + 5\) to get 15.

Step 3: Write simplified expression

Now write down the simplified expression. \(15x^{3}\) is the final, simplified form of our initial algebraic expression.

Key Concepts

Combining Like Terms

Understanding how to combine like terms is crucial in simplifying algebraic expressions. Like terms, as the name suggests, are terms that share the same variables raised to the same power. Think of them as matching puzzle pieces that can fit together perfectly. For example, in the exercise with expression 10x^3+5x^3, both terms have the variable x raised to the third power. Combining these like terms is straightforward: you add their coefficients, the numerical parts, and keep the common variable with its exponent unchanged.

To practice this concept, consider the expression 2a+3a. They're like terms because they both have the variable a. To combine them, add 2+3 to get 5a. It's important not to change the variable part when combining; the expression 2a+3b, for example, can't be combined into a single term because the variables differ.

Algebraic Coefficients

Algebraic coefficients are the numbers that multiply the variables in an algebraic expression. They are the anchors that give the variables weight and quantity. When simplifying expressions like the one in our exercise, 10x^3+5x^3, the coefficients are 10 and 5. These coefficients determine how heavily the variable x is counted in each term.

Understanding Coefficients with a Real-Life Analogy

Think of algebraic coefficients as ingredients in a recipe—say, cups of flour in a cake. If you have 3 cups in one bowl and 2 cups in another, combining them gives you 5 cups of flour. Similarly, when you add the coefficients 10 and 5, you end up with a total 'weight' of 15 for the variable x. In short, coefficients tell you how many 'parts' of the variable you have.

Exponents in Algebra

Exponents in algebra indicate how many times a number or variable is multiplied by itself. They are compact notations that can simplify complex multiplication. For instance, in our exercise, the exponent 3 in x^3 tells us that x is multiplied by itself three times (x*x*x).

Exponents remain unchanged when combining like terms as long as the bases (the variables or numbers getting multiplied) and the exponents are identical. If the exponents or bases differ, the terms are not like terms and cannot be combined this way. Thus, x^3 and y^3 cannot be combined, nor can x^2 and x^3, since in both cases either the bases or the exponents are different. Remember, when simplifying, it’s the coefficients that get combined, not the exponents.