Problem 15 - Fundamentals of Mathematics

Question

Write the expressions using exponential notation. 15 $$4 \cdot 4$$

Step-by-Step Solution

Answer

The expression \(4 \times 4\) is \(4^2\) in exponential notation.

Step 1: Understanding the Problem

We are given the expression \(4 \times 4\) and need to express this using exponential notation.

Step 2: Identify the Base and Exponent

In exponential notation, the base is the number being multiplied by itself. Here, the base is 4, and since it's multiplied by itself once, the exponent is 2.

Step 3: Write the Expression in Exponential Notation

Using exponential notation, the expression \(4 \times 4\) is written as \(4^2\). This means 4 raised to the power of 2.

Step 4: Final Answer

The expression \(4 \times 4\) in exponential notation is \(4^2\).

Key Concepts

Base and Exponent

The concept of a base and an exponent is central to understanding exponential notation. When you have repeated multiplication of the same number, you can simplify it using what we call exponential notation. Here, the base is the number that is being multiplied. Think of it as the main character in a story, the number doing all the heavy lifting!

• The base in our expression is 4.
• The exponent tells us how many times the base is used as a factor.
• In our case, because 4 is multiplied by itself, we use an exponent of 2.

In mathematical terms, this is noted as \(4^2\), which is read as "4 squared." Exponents make complex multiplication simpler and more organized, displaying a lot of information in a very compact form.

Multiplication

Multiplication is one of the fundamental operations in mathematics. It's often described as repeated addition. This means you're adding the same number several times. Say you have \(4 \times 4\), this means you are essentially adding four 4’s together: \(4 + 4 = 16\).

• Multiplication quickens the process of addition.
• It allows for more organized and efficient calculations.
• Especially useful in algebra for simplifying expressions.

In our exercise, multiplication of 4 by itself sets the stage for using exponential notation. It ties directly to how often the base, 4, appears, leading us to use an exponent of 2 to signify this repeated action.

Mathematical Expressions

Mathematical expressions are combinations of numbers and operations that represent values or relationships. One of the key aspects of expressions is their ability to convey complex ideas succinctly. The expression \(4 \times 4\) can be transformed into \(4^2\), making it not only more concise but also opening doors to further mathematical exploration like algebra.

• Expressions can include numbers, variables, and operators.
• They can be simplified, expanded, or rewritten for various purposes.
• Using exponential notation is one way to simplify multiplication expressions.

Understanding how to read and write expressions in different forms, such as exponential notation, enriches your mathematical literacy and problem-solving toolkit. This transforms longer, more cumbersome calculations into manageable, easily understandable pieces.