Problem 27 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. $$ 4 x 4 x 4 x 4 x 4 x $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(4 \times 4 \times 4 \times 4 \times 4\) in exponential notation. Answer: \(4^5\)

Step 1: Identify the base and count the number of repetitions

Identify the base (the number being multiplied) and count how many times it is repeated in the expression. In this expression, the base is 4, and it is being multiplied by itself 5 times.

Step 2: Write the expression in exponential notation

Write the expression using the power rule, which states that a number multiplied by itself n times can be written as the number raised to the power n. Since the base is 4 and it's being multiplied 5 times, write the expression in exponential notation as follows: $$ 4^5 $$

Step 3: Confirm the result

Verify that the exponential notation, \(4^5\), represents the original expression. According to the power rule, the given expression represents \(4 \times 4 \times 4 \times 4 \times 4\), which is the same as the original expression. Therefore, the exponential notation of the given expression is correct.

Key Concepts

Base and Exponent

In mathematics, the terms 'base' and 'exponent' are fundamental when dealing with exponential expressions. When you see an expression like \(4^5\), 4 is known as the 'base', and 5 is the 'exponent'.
Understanding these terms is crucial as they define how the expression is constructed.

• The base is the number that is being multiplied.
• The exponent tells us how many times the base is used as a factor.

Take for example the expression \(4^5\). Here, 4 is the base, meaning it is the number we are repeatedly multiplying. The 5 is the exponent, indicating that 4 is used as a factor five times. This makes it easier to express long multiplication in a more compact form.

Repeated Multiplication

Repeated multiplication is the process of multiplying the same number by itself several times.
This concept is what exponential notation is built upon.
Imagine you are tasked with multiplying 4 by itself five times, like this: \[4 \times 4 \times 4 \times 4 \times 4\]Writing this out in full every time can get cumbersome, which is why we use exponential notation to simplify it.

• Instead of writing multiple numbers again and again, exponential notation lets us write it in a shorthand form.
• Repetition is indicated by writing the base and the exponent.

So the expression \(4^5\) is much simpler than writing it out fully. It also makes calculations easier to manage, particularly in more complex mathematical problems.

Power Rule

The power rule in mathematics gives a method to write expressions in a compact form using exponents.
It is a shortcut that simplifies the representation of repeated multiplication of the same base number. According to the power rule:

• If you have a number \(b\) multiplied by itself \(n\) times, it can be written as \(b^n\).

For example, using the power rule, \[4 \times 4 \times 4 \times 4 \times 4 \]can be elegantly expressed as \(4^5\). This is essential not only for simplification but also serves as the foundation for more advanced mathematics.
Understanding and applying the power rule helps streamline calculations and can reduce errors when working with repetitive multiplications, hence providing a clearer view of the mathematics involved.