Problem 5 - Fundamentals of Mathematics

Question

Write each number without exponents. \(4^{7}\)

Step-by-Step Solution

Answer

The number \(4^7\) written without exponents is \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\).

Step 1: Understand the Problem

The problem states that we need to write the given number, which is presented as an exponent, without using exponents. The number is given as \(4^7\). This means that the number 4, known as the base, is multiplied by itself 7 times.

Step 2: Multiply the Base

To write \(4^7\) without exponents, you need to multiply the base 4 by itself 7 times. This looks like: \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\).

Step 3: Write the Expanded Form

Simply list out the multiplication as described in the previous step. The expression \(4^7\) can be expanded to: \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\).

Key Concepts

Exponentiation

Exponentiation is a powerful mathematical operation that allows us to express repeated multiplication in a compact form. When using exponentiation, we have two main components: the base and the exponent. The base represents the number that is being multiplied repeatedly, and the exponent indicates how many times it is being multiplied by itself.
For instance, in the expression \(4^7\), 4 is the base, and 7 is the exponent. This tells us that the number 4 is multiplied by itself 7 times. Exponentiation provides a simplified way to handle large numbers that would otherwise be cumbersome to write out fully, making it a key concept in mathematics.
In everyday scenarios, exponentiation is often used in computing powers of numbers, calculating scientific calculations, and even analyzing exponential growth in populations or investments.

Base and Exponent

In the operation of exponentiation, understanding the terms 'base' and 'exponent' is essential. The base is the number subjected to repeated multiplication, while the exponent tells you how many times the base is used in the multiplication process.
If we consider the expression \(4^7\):

• Base: The number 4 is the base. It is the number being multiplied.
• Exponent: The number 7 is the exponent. It tells us that the base (4) should be multiplied by itself a total of 7 times.

By using both base and exponent together, we can quickly identify how many times a number is intended to be multiplied, providing clarity and structure to mathematical operations.
This specific terminology is fundamental in both mathematics and sciences, as it ensures precision and avoids misunderstandings in complex calculations.

Multiplication as Repeated Addition

At its core, multiplication is a form of repeated addition, which is a concept that stems from basic arithmetic. When you multiply a number, you are essentially adding the same number multiple times. Exponentiation takes this concept a step further by allowing repeated multiplication rather than repeated addition.
For example, to compute \(4^7\) using the principles of multiplication as repeated addition, you would consider:

• First step: \(4 \times 1 = 4\)
• Second step: \(4 + 4 = 8\), hence \(4 \times 2 = 8\)
• Continuing this process to \(4 \times 7 = 28\)

Now, when dealing with exponents, rather than adding, you are multiplying repeated times:
\[4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\]
This delineates the growth from simple addition and multiplication towards more abstract calculations, showing how powerful tools like exponents become essential when handling larger numbers or more complex problems.