Problem 22 - Fundamentals of Mathematics

Question

Write the expressions using exponential notation. \(\underbrace{2 \cdot 2 \cdots \cdots 2}_{112 \text { factors of } 2}\)

Step-by-Step Solution

Answer

The expression is written as \(2^{112}\).

Step 1: Understanding the Problem

We are given a long multiplication of the number 2 repeated 112 times. This can be expressed in exponential form, as repeated multiplication of the same number is represented by exponents.

Step 2: Identify the Base

In the expression \(2 \cdot 2 \cdot \ldots \cdot 2\), every factor is the number 2. Therefore, our base in the exponential expression is 2.

Step 3: Count the Number of Factors

The problem states there are 112 factors of 2 in the expression. This count is what will become our exponent in the exponential notation.

Step 4: Write in Exponential Notation

The expression with 112 factors of 2 can be written as \(2^{112}\). The base is 2 and the exponent is 112, which indicates how many times the number 2 is multiplied by itself.

Key Concepts

Base and Exponent

In exponential notation, we often see a small number written to the top right of a larger number. This is a powerful way to show that a number, called the "base," is multiplied by itself a certain number of times. The smaller number is called the "exponent." Let's break this down further:

• Base: The base is the number that is being multiplied. In the expression \(2^{112}\), the base is 2. It is the number that repeatedly appears in the multiplication sequence.
• Exponent: The exponent tells us how many times the base multiplies itself. For \(2^{112}\), the exponent is 112. This means 2 is used as a factor 112 times in the repeated multiplication process.

Using both a base and an exponent simplifies large multiplication tasks. Rather than writing out dozens or even hundreds of multiplications, you can represent the entire operation in a compact form like \(2^{112}\). This notation makes it easier to understand and work with large computations in mathematics.

Repeated Multiplication

The concept of repeated multiplication is fundamental to understanding exponential notation. When we talk about repeated multiplication, we mean multiplying the same number by itself multiple times. For example, in the original exercise, the number 2 is multiplied by itself 112 times.In simpler terms, imagine you have to multiply 2 by itself over and over again:

• Once: \(2\)
• Twice: \(2 \cdot 2 = 2^2\)
• Three times: \(2 \cdot 2 \cdot 2 = 2^3\)
• ...and so on, until 112 times: \(2^{112}\)

Instead of writing 112 twos and multiplying them, exponential notation lets us express this as \(2^{112}\). This is particularly useful when dealing with very large numbers of factors, which would be cumbersome to handle otherwise.

Expressing Powers

When we use exponential notation, we are expressing powers. A "power" is just another way of saying repeated multiplication of a base by itself, indicated by the exponent. The original exercise tasked us with taking a long string of multiplications and turning it into an exponential expression. Here's how that works in practice:

• Start with the base, which is the number being multiplied. In our example, that's 2.
• Use the exponent to show how many times the multiplication occurs. So, with 2 multiplied by itself 112 times, we express that as 2 raised to the power of 112, or \(2^{112}\).

Expressing numbers as powers, such as \(2^{112}\), allows us to work more efficiently with large numbers. It simplifies calculations and makes mathematical expressions more manageable and easier to read. This process of using powers becomes increasingly important in advanced mathematics and helps lay the groundwork for higher-level concepts like logarithms and scientific notation.