Problem 17 - Algebra 1: Concepts and Skills

Question

Write the expression in exponential form. \(3 \cdot 3 \cdot 3 \cdot 3\)

Step-by-Step Solution

Answer

The expression \(3 \cdot 3 \cdot 3 \cdot 3\) can be written in exponential form as \(3^4\).

Step 1: Identify the Base

The base number is the number being repeatedly multiplied. In this case, the base number is \(3\).

Step 2: Count the Occurrences of the Base

Count how many times the base number is being multiplied. Here, \(3\) is being multiplied four times.

Step 3: Write the Expression in Exponential Form

Write the base number with an exponent that's equal to the number of times the base is being multiplied. In this case, \(3 \cdot 3 \cdot 3 \cdot 3\) can be written as \(3^4\).

Key Concepts

Exponential Form

Exponential form is a way to express numbers through powers. It simplifies repeated multiplication of the same number. Instead of writing a number several times, like in the repeated multiplication of 3 (i.e., \(3 \cdot 3 \cdot 3 \cdot 3\)), we use exponential form. This allows us to write it as \(3^4\).

• The number being multiplied is the "base number."
• The little number above and to the right is the "exponent."
• The exponent tells you how many times to multiply the base by itself.

In our exercise, writing \(3 \cdot 3 \cdot 3 \cdot 3\) in exponential form gives us \(3^4\). This method reduces complexity and is a valuable skill, especially when dealing with larger numbers.

Base Number

The base number is the main number in any exponential expression. It is the number that gets multiplied repeatedly. In the example \(3^4\), the base number is 3. This means 3 is multiplied by itself a total of four times.

• Understand that the base itself must be a number with potential for repetition.
• It acts as the foundation for the expression.
• In exponential expressions, we mainly focus on outcomes after the base is used to multiply.

The base number remains constant in multiple calculations, allowing for efficient scalability when managing repeated calculations.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It represents the process of multiplying the base number by itself as many times as indicated by the exponent.

• The base is multiplied repeatedly, controlled by the exponent.
• It is a key concept in mathematics, used in many fields such as algebra and calculus.
• Exponentiation helps in expressing large numbers more compactly and managing them efficiently.

In our scenario, exponentiation turned \(3 \cdot 3 \cdot 3 \cdot 3\) into a compact \(3^4\), making calculations more digestible and comprehension more straightforward. Understanding exponentiation opens doors to mastering mathematical expressions at a larger scale.