Problem 1 - Precalculus Mathematics for Calculus

Question

(a) Using exponential notation, we can write the product \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \text { as }\) _____________. (b) In the expression \(3^{4},\) the number 3 is called the ____________, and the number 4 is called the ___________.

Step-by-Step Solution

Answer

(a) \(5^6\); (b) Base, exponent.

Step 1: Understanding Exponential Notation

Exponential notation is a way of expressing repeated multiplication of the same factor. For example, when you multiply a number by itself several times, you can write it using a base number and an exponent. In this case, the base is the number being multiplied, and the exponent indicates how many times the base is used as a factor.

Step 2: Converting Product to Exponential Notation

To express the product \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\) in exponential notation, first identify that 5 is being multiplied by itself a total of 6 times. The base number here is 5, and the exponent represents the number of times the base appears in the multiplication. Therefore, this expression can be written as \(5^6\).

Step 3: Identifying Components of an Exponential Expression

In any exponential expression like \(3^4\), the base of the expression is the number being repeatedly multiplied, which in this case is 3. The exponent, which indicates how many times the base is used as a factor, is 4 in this expression.

Key Concepts

Base Number

The concept of a base number is fundamental in understanding exponential notation. When we talk about exponential expressions, the base number is the one that is being repeatedly multiplied. It holds the key position in the notation structure. For instance, in the expression \(5^6\), the number 5 is the base number. It is the 'factor' that you see repeated in the multiplication chain.
Think of the base number as the building block of repeated multiplication. It tells us "which number" is at play. In any exponential expression:

• The base number is the "bottom" number, or the larger, initially visible number.
• It is the primary focus of the multiplication process.
• It shows up as many times as indicated by the exponent in the multiplication.

Understanding the base number is crucial because recognizing it allows you to quickly identify what is being done repeatedly, without listing it out each time.

Exponent

An exponent is a superscript number found in exponential notation that dictates the number of times the base number should be multiplied by itself. In simple terms, it's a shorthand way to express repeated multiplication. For example, in \(3^4\), the number 4 is the exponent, showing that 3 should be multiplied by itself four times: \(3 \times 3 \times 3 \times 3\).
Here's why the exponent is important:

• It simplifies expressions by reducing lengthy multiplication steps into a concise form.
• The exponent tells you "how many times" the base number is used as a factor.
• It is always written slightly above and to the right of the base number in an exponent expression.

This concept helps not just in simplifying multiplication but also in understanding powers and roots in mathematics. Recognizing the exponent's role provides clarity in more complex calculations and in interpreting scientific and mathematical formulas.

Repeated Multiplication

Repeated multiplication forms the core of exponential notation. When we express numbers in this format, we're essentially using a shorthand for multiplying the same number over and over. This approach helps save time and space, especially as numbers get larger. For example, instead of writing out \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\), expressing it as \(5^6\) conveys the same information with much less clutter.
Repeated multiplication is useful for:

• Streamlining complex calculations.
• Making equations easier to read and write.
• Understanding growth patterns, particularly in geometric sequences and scientific contexts.

By grasping this concept, you can handle power terms quickly and efficiently in both everyday math problems and higher-level computations. It forms a basis for understanding other mathematical phenomena, like compound interest and exponential decay, found in various real-world applications.