Problem 2 - Fundamentals of Mathematics

Question

Write the following using exponents. \(16 \cdot 16 \cdot 16 \cdot 16 \cdot 16\)

Step-by-Step Solution

Answer

\(16^5\)

Step 1: Identify the Repeated Base

First, we need to identify the number that is being multiplied repeatedly. In this case, the base number is 16.

Step 2: Count the Multiplications

Next, we count how many times the base number 16 is used in the multiplication. Here, 16 is used 5 times.

Step 3: Write Using Exponents

Using the base and the count from the previous steps, we express the repeated multiplication in exponential form. Since the base is 16 and it is used 5 times, it can be written as \(16^5\).

Key Concepts

Base Number

The base number is a key element in understanding exponents, and it refers to the number that is multiplied by itself repeatedly. In our exercise, the base number is 16. It's the number that appears multiple times in the expression. To identify the base number, simply look at the repeated factor in a multiplication expression. Once identified, the base helps us to rewrite multiple multiplications in a more compact form. It's important to remember: - The base can be any number: whole numbers, fractions, and even negative numbers. - Recognizing the base number helps simplify multiplication problems by using exponents.

Exponential Form

Exponential form is a way to represent repeated multiplication in a concise manner using exponents. It is made up of two parts: the base number and the exponent.The **base number** is the factor that is repeated in the multiplication, while the **exponent** tells us how many times to multiply the base by itself.Exponential form is written as:- \[\text{{base}}^{\text{{exponent}}}\] - In this notation, the base is the number being repeatedly multiplied, and the exponent is the number of times it appears as a factor.For example, in the expression \(16^5\), 16 is the base, and 5 is the exponent, indicating that 16 is used as a factor in the multiplication 5 times. This form makes calculations simpler and equations more manageable.

Multiplication

Multiplication is one of the fundamental operations in mathematics and involves adding a number to itself a certain number of times. It becomes simpler and more manageable when we use exponents.When the same number is multiplied repeatedly, it's efficient to express it in a shorter form using exponents. In this way:

• The base number is the one being multiplied.
• The exponent shows how many times the multiplication occurs.

For instance, instead of writing out \(16 \cdot 16 \cdot 16 \cdot 16 \cdot 16\), we use exponential form: \(16^5\).This method helps:- Streamline complex problems by reducing the number of steps needed in calculations.- Simplify the representation and handling of very large numbers in equations.