Problem 82 - Prealgebra

Question

Perform the indicated operations. $$8^{2}$$

Step-by-Step Solution

Answer

The result is 64.

Step 1: Understanding the Exponentiation

Exponentiation means multiplying the base by itself as many times as the exponent indicates. In this case, we have the base as 8 and the exponent as 2.

Step 2: Performing the Multiplication

We need to multiply the base 8 by itself because the exponent is 2. So, calculate the following: \[8 \times 8\].

Step 3: Calculating the Result

Perform the multiplication: \[8 \times 8 = 64\]. Therefore, the value of \(8^{2}\) is 64.

Key Concepts

Base and Exponent

In mathematics, exponentiation plays a crucial role in simplifying repeated multiplication. To understand this, we focus on two key components: the base and the exponent.

• The **base** is the number you want to multiply by itself. In our example, the base is 8.
• The **exponent** tells you how many times to multiply the base by itself. Here, the exponent is 2.

Imagine the base as the foundation and the exponent as the number of stories, or levels, you want to build. In simpler terms, for a base like 8 raised to the power of 2, you are tasked to build a structure with two stories using the number 8. A powerful concept, it allows much more concise expressions of large multiplication problems.

Multiplication

Once you understand the base and exponent, the next step is multiplication. This operation is about repeated addition.
For exponentiation, you multiply the base number repeatedly by itself according to the number of times the exponent specifies.
Taking our example of \[8^{2}\], you multiply 8 by itself:
\[8 \times 8\]
This multiplication process is straightforward but foundational to achieving the exponentiation outcome correctly.Each multiplication step is like laying bricks for a sturdy wall: precise and repetitive. It helps in understanding how numbers grow exponentially.

Calculating Powers

Calculating powers involves completing the multiplication to find the final result. It is the ultimate step in exponentiation where you see the result of your earlier steps.
For example, calculating \[8^{2}\] involves finding out how many instances of 8 multiplied together make up the final number:

• Start with your expression: \[8 \times 8\]
• Perform the multiplication: \[8 \times 8 = 64\]

So, et voilà, \(8^{2} = 64\).
Calculating powers effectively translates complex-looking expressions into understandable numbers. It reveals the large impact of small numbers when combined through multiplication.