Problem 48 - Algebra 1

Question

Write the expression in exponential form. \(15 \cdot 15 \cdot 15 \cdot 15\)

Step-by-Step Solution

Answer

In exponential form, the expression \(15 \cdot 15 \cdot 15 \cdot 15\) can be written as \(15^4\).

Step 1: Identify the Base

In this expression, the number that is being multiplied repetitively is \(15\). This number is the base in an exponential expression. So, the base is \(15\).

Step 2: Count the number of times the base is multiplied

The number of times the base is being multiplied is called the exponent in an exponential expression. In this expression, the base \(15\) is being multiplied \(4\) times.

Step 3: Write the expression in exponential form

The exponential form of an expression is written as \(base^{exponent}\). Using the base and exponent derived from the previous steps, the expression can be rewritten in exponential form as \(15^4\).

Key Concepts

Understanding the Exponent

The concept of an exponent can seem daunting at first. But it's actually quite simple once you break it down.

The exponent is the small number located at the upper right of a base number. It indicates how many times the base number is multiplied by itself. This is like shorthand for multiplication. Instead of writing a number over and over, you use the exponent to show repetition in a compact form.

• Think of the exponent as a multiplier of time. It tells you the frequency of multiplication, not the value itself.
• For example, in the expression, \(2^3\), the exponent is \(3\) and it tells you that the base 2 is multiplied by itself three times, which results in \(2 \cdot 2 \cdot 2 = 8\).

Exploring the exponential form of expressions reveals the beauty and efficiency of using exponents. Always remember, the larger the exponent, the more times you'll multiply the base number by itself.

The Role of the Base Number

In any exponential expression, the base number holds fundamental importance. It is the number that serves as the backbone of the expression.

The base number is the main figure that you multiply repeatedly. When represented in exponential form, it is the larger number in front of the exponent.

• For instance, in \( 5^3 \), the base number is \(5\).
• The base determines the output value when multiplied by itself according to the exponent.

Understanding the base number helps you visualize what the exponent is acting upon. This is essential in interpreting and creating exponential expressions. Whenever you are asked to write a repeated multiplication in exponential form, identifying the base number is your first step.

Crafting an Exponential Expression

An exponential expression is a way of representing repeated multiplication in a simplified and concise format. This comes into play when a number is used several times in multiplication.

Given a problem like \(15 \cdot 15 \cdot 15 \cdot 15\), you can express it exponentially.

• First, identify the base number. Here, it's \(15\), as it is the number being multiplied repeatedly.
• Next, determine the exponent by counting how many times the base appears as a factor. In this example, \(15\) appears four times. Thus, the exponent is \(4\).
• Combine the base number and exponent to write the expression in exponential form: \(15^4\).

Exponential expressions are powerful tools in mathematics, allowing for efficient computation and easier handling of large numbers in repeated multiplication scenarios.