Problem 34 - Algebra 1

Question

Evaluate the power. $$ 2^{5} $$

Step-by-Step Solution

Answer

The value of \(2^{5}\) is 32.

Step 1: Understanding the notation

The problem gives a number in power format. The number 2 is the base, and 5 is the exponent. This means we multiply the base number by itself, the number of times indicated by the power.

Step 2: Computing the power

We then perform the multiplication \(2 \times 2 \times 2 \times 2 \times 2\).

Key Concepts

Understanding Base and Exponent

When dealing with exponents, two key components are the base and the exponent. Let's dive into these terms to understand them better.
- **Base**: This is the number that you multiply by itself. In our example, the number 2 is the base. - **Exponent**: This signals how many times you should multiply the base by itself. In this context, the number 5 indicates that the base 2 will be multiplied by itself five times.
Understanding these two components is crucial. They tell you exactly what calculations you need to perform. They are always written together in a specific order, with the base located at the bottom and the exponent as a small number to its top right.

Exploring Power Notation

Power notation is a compact way of expressing repeated multiplication of a number by itself.
In power notation, the expression is written as **base^exponent**. For example, in the expression \(2^5\): - **Base (2)** tells us the number we start with.- **Exponent (5)** provides the instructions to multiply the base five times.This notation is not just for convenience—it's very practical. By writing \(2^5\), we save space and avoid long, repetitive expressions that can become cumbersome.
Power notation is found in various areas of math and science, making it a fundamental tool for expressing larger calculations succinctly.

Simplifying with Multiplication

Multiplication is at the core of what exponents do. When you see a power such as \(2^5\), it instructs you to multiply 2 by itself repeatedly. Let's see how you carry out this multiplication step-by-step.
1. Start with the base, 2.2. Multiply by 2 again. You have now done \(2 imes 2\).3. Continue to multiply by 2 as specified by the exponent: - Third multiplication: \(2 imes 2 imes 2\)
- Fourth multiplication: \(2 imes 2 imes 2 imes 2\)
- Fifth multiplication: \(2 imes 2 imes 2 imes 2 imes 2\)
By following these sequential steps, you use the rules of basic arithmetic. The result will give you the accurate value of the power notation, in this case, 32. Remember, each multiplication is simply building upon the last, relying on a solid understanding of how multiplication works.