Problem 1 - Algebra and Trigonometry

Question

Using exponential notation, we can write the product 5\(\cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\) as _____.

Step-by-Step Solution

Answer

The product is written as \(5^6\).

Step 1: Identify the Repeated Multiplication

First, observe the expression to identify what numbers are being multiplied repeatedly. Here, you have the numbers \(5\) being multiplied by itself.

Step 2: Count the Number of Repetitions

Next, count how many times the base number \(5\) is repeated in the multiplication. You see the number \(5\) repeated \(6\) times.

Step 3: Write in Exponential Form

Now, write the expression using exponential notation. Since the base \(5\) is repeated \(6\) times, we express it as \(5^6\).

Step 4: Verify the Solution

Check that the expression \(5^6\) accurately represents the original repeated multiplication \(5\cdot 5 \cdot 5 \cdot 5\cdot 5 \cdot 5\). Each instance of the number \(5\) corresponds to an exponent count.

Key Concepts

Repeated Multiplication

Repeated multiplication involves multiplying the same number over and over. It's like having a collection of the same factor repeated in a single multiplication operation. For instance, when you see the expression \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\), it's a way of saying you are multiplying the number \(5\) by itself repeatedly.
This pattern of repeated multiplication can easily get confusing with large numbers or many repeats. That's where having a simpler notation, like exponential notation, becomes helpful.

Exponent

In mathematics, an exponent represents how many times a number, known as the base, is used as a factor in repeated multiplication. It is written as a small number to the right and above the base number. For example, in \(5^6\), the \(6\) is the exponent. This tells us that the number \(5\) is multiplied by itself a total of six times.
Exponents help to simplify expressions by reducing long chains of repeated multiplication into more compact and manageable forms. This can make computation easier and solutions more elegant.

Base Number

The base number in exponential notation refers to the number that is repeatedly multiplied. It's the foundation of the repeated multiplication before any exponents are involved. In our exercise, the number \(5\) is the base number.
Understanding the base number's role in expressions allows scientists, mathematicians, and students to decipher and create meaningful representations of complex calculations. The base simply shows what number you start with before you apply the exponent. Knowing this is crucial for deciphering and solving mathematical problems involving exponents.