Problem 16 - Fundamentals of Mathematics

Question

Write the expressions using exponential notation. $$12 \cdot 12$$

Step-by-Step Solution

Answer

\(12^2\)

Step 1: Identify the Base Number

In exponential notation, we need to identify the base number and the number of times it is multiplied by itself. In this expression, we have two 12s being multiplied together, so the base number is 12.

Step 2: Determine the Exponent

The exponent tells us how many times the base number is used as a factor. Since we have two 12s being multiplied, the exponent is 2.

Step 3: Write in Exponential Notation

Using exponential notation, the expression is written as \(12^2\), indicating that 12 is multiplied by itself once.

Key Concepts

Base Number

The base number in exponential notation is the number that acts as the foundation for the operation. It is the value that is repeatedly multiplied by itself. In the expression given, \(12 \cdot 12\), the base number is 12. This means that 12 is the number upon which the exponent acts.

To identify the base in similar mathematical expressions, look for the number that occurs repeatedly.

• If the expression was \( ext{9} \times ext{9} \times ext{9} \), then the base number would be 9.
• In the expression \( ext{5}^{3} \), the number 5 is already identified as the base.

Always remember, the base is the starting point of multiplication when dealing with exponential notation.

Grasping this component is key, as it determines the entire operation of the expression.

Exponent

The exponent is an essential part of exponential notation. This small superscript number indicates how many times the base number is utilized as a factor in the multiplication process.

For the expression \(12 \cdot 12\), the exponent is 2, because 12 is multiplied by itself one time. In mathematical notation, this is represented as \(12^2\).

Here are some important aspects to remember about exponents:

• The exponent ‘1’ means the base number is used as a factor once, just as itself.
• If the exponent were ‘3’, for example, it implies the base is multiplied together three times, like \(12 \cdot 12 \cdot 12\).

Consistently determining the exponent allows clarity in simplifying and handling complex expressions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like addition, subtraction, multiplication, and division) that collectively define a quantity. In the task you are dealing with, these expressions are simplified using exponential notation.

The expression \(12 \cdot 12\) is simplified to \(12^2\). This helps in both reducing the complexity of calculations and communicating mathematical ideas clearly.

• Expressions with exponents are less cumbersome than their expanded form.
• They are particularly useful in indicating repeated multiplication succinctly.

Employing exponential notation in mathematical expressions can make complex operations easier to manage, helping you solve problems more efficiently. Understanding how to transform regular expressions into exponential form is a vital mathematical skill.