Problem 17 - Fundamentals of Mathematics

Question

Write the expressions using exponential notation. $$9 \cdot 9 \cdot 9 \cdot 9$$

Step-by-Step Solution

Answer

The expression is \(9^4\).

Step 1: Identify the base

First, observe that each number in the given expression is 9. This means that the repeated factor, known as the base of the exponent, is 9.

Step 2: Count the number of repeated factors

Next, count how many times the base 9 is repeated in the multiplication. Here, 9 appears 4 times.

Step 3: Write the expression using exponential notation

Combine the base and the number of times it is repeated, and write the expression in exponential notation as \(9^4\). This means that 9 is multiplied by itself 4 times.

Key Concepts

Base of Exponent

In exponential notation, the **base** plays a crucial role. It represents the number that is being repeatedly multiplied. For instance, in the expression \(9 \cdot 9 \cdot 9 \cdot 9\), all the numbers are the same, specifically 9. This number 9 forms the base when we convert this multiplication into exponential notation. The base is always the factor that is multiplied multiple times in the expression.

Understanding the base helps you grasp what number is "piled upon" itself in multiplication. It essentially forms the foundation upon which exponential notation is built. Keep in mind:

• The base is the continually repeated number.
• It is the "root" of the expression in exponential form.
• In \(9^4\), 9 is the base.

Repeated Factors

The concept of **repeated factors** is pivotal in understanding exponential notation. A repeated factor is a number that appears multiple times in a multiplication sequence. Recognizing these recurring numbers allows you to transition from lengthy multiplication to concise exponential expressions.

For example, in multiplying \(9 \cdot 9 \cdot 9 \cdot 9\), the number 9 is the repeated factor, as it appears 4 times. By identifying repeated factors, you simplify calculations and clarify understanding of the expression's structure. This is the first step in converting a multiplication sequence into exponential form. Remember:

• Repeated factors show how many times a number appears.
• They distinguish the base in exponential notation.
• In \(9^4\), "4" indicates the number of repetitions of the base 9.

Multiplication in Exponents

**Multiplication in exponents** is a method to simplify multiple identical factors through exponential notation. Rather than writing an extensive multiplication expression, exponents allow for a more streamlined approach.

In our case, \(9 \cdot 9 \cdot 9 \cdot 9\) is simplified to \(9^4\). Here, 9 is multiplied by itself 4 times, which is succinctly expressed as an exponent, where 9 is the base and 4 is the exponent. This method not only makes expressions easier to understand but also speeds up calculations in more complex mathematical problems. Key points:

• Exponential notation reduces repeated multiplication.
• The exponent shows how many times to multiply the base.
• It provides clarity and efficiency in mathematical expressions.