Question

Rewrite \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) with an exponent. A. \(2^5\) B. \(5^2\) C. 32 D. 10

Short answer

A. \(2^5\)

Step-by-step solution

Identify the base number

Observe the repeated multiplication: 2 is the number being multiplied repeatedly.

Count the number of multiplications

Count how many times the number 2 is being multiplied: 2 is multiplied 5 times.

Rewrite using an exponent

Using exponents, repeated multiplication of a number can be expressed as the base raised to the power of the number of times it is multiplied. Therefore, rewrite the expression as: \(2^5\)

Select the correct answer

Compare the rewritten expression \(2^5\) to the given multiple-choice options: A. \(2^5\) B. \(5^2\) C. 32 D. 10The correct answer is option A.

Key concepts

Base Number

In mathematics, a base number is the number that is used as the factor in repeated multiplication when dealing with exponents.
When we see an expression like \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\), the number 2 serves as the base number.
This is because it is the number that is being multiplied repeatedly.
The concept of a base number is crucial for understanding how exponents work and how to simplify expressions.
In an exponential expression, the base number is what gets raised to a power, or exponent. For example, in \(2^5\), 2 is the base number that is multiplied by itself five times.

Repeated Multiplication

Repeated multiplication is simply the process of multiplying the same number several times.
For instance, when you multiply 2 by itself five times, you get \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\).
This can be quite tedious to write out for large numbers, and that's where exponents come in handy.

Instead of writing out 2 multiplied by itself five times, we can use an exponent to simplify this expression. By recognizing that multiplication is repeated, we can use the base number and count how many times it is repeated to form an exponential expression.
For example, \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) can be rewritten as \(2^5\), where 2 is the base number and 5 is the exponent.

Exponential Notation

Exponential notation is a way to express repeated multiplication of the same number in a compact form.
It involves a base number and an exponent. The exponent tells us how many times to multiply the base number by itself.

In our example, the notation \(2^5\) tells us that the base number 2 is multiplied by itself five times.
This is much simpler than writing out all the multiplications, especially for larger numbers.
Exponential notation is essential in higher-level math and real-world applications, such as scientific calculations and computer science.
Understanding how to convert between repeated multiplication and exponential notation will make solving math problems quicker and more efficient.