Problem 33 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(10^{2}\)

Step-by-Step Solution

Answer

The value of \(10^2\) is 100.

Step 1: Understand the Problem

We need to calculate the value of the expression \(10^2\), where 10 is the base and 2 is the exponent. This means we are looking for the number obtained when multiplying the base (10) by itself as many times as indicated by the exponent (2).

Step 2: Calculate the Power

Since \(10^2\) means 10 multiplied by itself once (as 2 indicates two occurrences), we perform the calculation: \(10 \times 10 = 100\).

Step 3: Verify with a Calculator

To ensure our manual calculation is correct, use a calculator and input \(10^2\). The calculator should confirm that \(10^2 = 100\).

Key Concepts

Base and Exponent

In mathematics, understanding the concept of base and exponent is crucial when dealing with exponents, often referred to as powers. In our expression, \(10^2\), 10 is known as the base, and 2 is the exponent. The base is the number that is repeatedly multiplied. The exponent tells us how many times the base is used in the multiplication.
For example:

• In the expression \(10^2\), "10" is the base.
• The "2" is the exponent, meaning 10 is multiplied by itself once. This results in the expression \(10 \times 10\).

Understanding this relationship helps simplify and solve problems quickly and efficiently. The process involves raising a number to the power indicated by the exponent, giving the power of a number.

Multiplication

Multiplication is a fundamental arithmetic operation that plays a vital role when working with exponents. In the exercise, we calculated \(10^2\). Essentially, this means multiplying the base, 10, by itself according to the exponent.
Here's a breakdown:

• The expression \(10^2\) means 10 is multiplied by itself.
• Performing the multiplication: \(10 \times 10 = 100\).

This operation is straightforward because multiplication is a repeated addition. In our example, you add 10 to itself, which results in 100. Recognizing how exponents simplify these repeated multiplications is important in math, allowing complex calculations to become manageable.

Calculator Verification

After manually calculating exponents, it's a good practice to verify your results using a calculator. This ensures accuracy and helps confirm your understanding of the concept. Using a calculator simplifies verification and can reinforce learning.
To verify \(10^2\) manually calculated as 100:

• Input the expression into the calculator: \(10^2\).
• The calculator will display the result, confirming that \(10^2 = 100\).

It's important to double-check calculations because errors can occur, especially in more complex problems. Calculator verification provides confidence and reliability in your mathematical solutions.