Problem 16 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. \(x^{3} \cdot x^{2}=x^{?}\)

Step-by-Step Solution

Answer

The short answer is \(x^{5}\).

Step 1: Identify the base and the exponents

In this equation, \(x\) is the base and 3 and 2 are the exponents. When multiplying two numbers with the same base, the exponents are added.

Step 2: Add the exponents

Add the exponents together. 3 + 2 equals 5.

Step 3: Write the answer

Write the base and the new exponent. The answer is \(x^{5}\).

Key Concepts

Multiplying Exponents

Understanding how to multiply exponents is fundamental when dealing with exponential expressions. The process becomes straightforward when the bases being multiplied are the same. According to the exponent rules, when multiplying two exponential terms with the same base, you simply keep the base and add the exponents together.

For example, let's take the expression \(x^3 \cdot x^2\). Both terms have the same base, which is \(x\). According to the rule, we add the exponents: 3 + 2. Hence, the solution becomes \(x^{3+2}\) or \(x^5\). This operation is crucial since it simplifies the calculations and allows for more complex equations to be solved easily. Additionally, understanding this rule facilitates the comprehension of more advanced mathematics, such as polynomial multiplication and algebraic functions.

Exponential Expressions

An exponential expression consists of a base raised to an exponent. The base is the number that is multiplied by itself, and the exponent indicates how many times the multiplication occurs. For instance, in the expression \(x^3\), \(x\) is the base and the number 3 is the exponent, showing that \(x\) is multiplied by itself three times (\(x \cdot x \cdot x\)).

These expressions are not just limited to numerical bases; they can include variables, algebraic terms, and even more complex structures. They play a significant role in various fields like finance, science, and engineering, where exponential growth or decay patterns are modeled. Therefore, a firm grasp of how to manipulate these expressions mathematically is vital. Clear comprehension enables students to tackle growth models, compute compound interest, and understand radioactive decay, among other applications.

Base and Exponent

At the heart of exponential expressions are two components: the base and the exponent. The base is the value that gets repeatedly multiplied, and the exponent denotes the number of times the base is used as a factor in the multiplication. In simpler terms, the exponent tells you how many times you use the base in a multiplication.

For instance, in the expression \(5^2\), 5 is the base, and 2 is the exponent. The expression means that 5 is multiplied by itself once: \(5 \cdot 5\). The role of the base and exponent becomes even more pronounced when working with variables, as seen in \(x^3\) or \(y^2\). Recognizing these components is not only useful for simple computations, but also when applying more complex mathematical operations such as exponent division, power of a power, or negative exponents, which follow different sets of rules. Grasping these concepts lays the foundation for higher-level mathematics, including calculus and beyond.