Problem 57 - Algebra 1: Concepts and Skills

Question

Write the expression in exponential form. \(4 \cdot 4 \cdot 4\)

Step-by-Step Solution

Answer

The exponential form of \(4 \cdot 4 \cdot 4\) is \(4^3\).

Step 1: Identify the base number

In the multiplication \(4 \cdot 4 \cdot 4\), the base number, which occurs repeatedly, is 4.

Step 2: Count the number of times the base number repeats

The base number 4 repeats three times in the multiplication.

Step 3: Write in exponential form

Since 4 is repeated three times, the multiplication can be written in exponential form as \(4^3\).

Key Concepts

Base Number

In mathematics, a base number refers to the fundamental value that is used as a starting point in operations like multiplication or exponentiation. When we look at the expression \(4 \cdot 4 \cdot 4\), the base number is 4.

The base number is important because it signifies the value being multiplied. In situations involving repeated multiplication, it is this number that gets multiplied by itself as many times as the exponent indicates.

In our example, the base number 4 is central, because it provides the basic "unit value" so to speak, that transforms based on the repeated actions (multiplications) performed on it. Noticing the base number is key to converting expressions to exponential form.

Exponent

An exponent tells us how many times the base number needs to be multiplied by itself. It essentially provides a shortcut for indicating repeated multiplication without having to write out each multiplication step.

For example, in the expression \(4 \cdot 4 \cdot 4\), once we recognize 4 as the base number, we count how many times it appears in succession. Here, 4 appears three times. Therefore, we write it in exponential form as \(4^3\).

The power or exponent 3 indicates "4 multiplied by itself three times." This concept helps us work with large numbers more efficiently as it reduces the steps needed in multiplication processes.

Repeated Multiplication

Repeated multiplication is a process where the same number is multiplied by itself multiple times. This approach can be simplified by using exponential notation.

For the expression \(4 \cdot 4 \cdot 4\), each instance of 4 represents a step in the multiplication process. Instead of writing out each multiplication operation, we use exponents.

This expression turns into \(4^3\), pithily expressing a repeated multiplication process in a compact form. Using exponential notation makes it easy for us to understand and solve problems quickly, thereby facilitating computational efficiency and clarity.