Problem 42 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(100^{2}\)

Step-by-Step Solution

Answer

The value of \(100^2\) is 10000.

Step 1: Understanding Powers

In mathematics, a power refers to how many times a number, known as the base, is multiplied by itself. The number of times the base is multiplied is indicated by the exponent.

Step 2: Identify Base and Exponent

In the expression \(100^2\), the base is 100, and the exponent is 2. This means we need to multiply 100 by itself.

Step 3: Calculate the Power

To find \(100^2\), multiply 100 by 100. This can be written as: \(100 \times 100 = 10000\).

Step 4: Verify with Calculator

Use a calculator to confirm the calculation: enter 100, use the exponent function, enter 2, and verify the result as 10000.

Key Concepts

Base and Exponent

When we talk about powers in mathematics, we engage with two important elements: the base and the exponent. Understanding these elements is crucial for working with powers. The **base** is the number that is being multiplied. In our example, the base is 100. The **exponent**, on the other hand, tells us how many times the base is used in the multiplication. In the expression \( 100^2 \), the exponent is 2. This means we multiply 100 by itself. To simplify, in mathematical expressions such as \( a^n \), \( a \) is the base and \( n \) is the exponent. The combined concept of base and exponent is crucial in simplifying and solving mathematical problems involving powers.

Multiplication

Multiplication plays a critical role in solving expressions involving powers. When we have a power, like \( 100^2 \), it is asking us to perform repeated multiplication. Specifically, you take the base number—in this case, 100—and multiply it by itself as many times as the exponent indicates. With \( 100^2 \), we multiply 100 by itself just once, as the exponent is 2.This multiplication process results in the value for that power, giving us:

• \( 100 \times 100 = 10,000 \)

Understanding this repeated multiplication makes solving powers intuitive and straightforward. Whenever you encounter an expression with a power, think of it as a shorthand for multiplication.

Calculator Verification

After manually solving an expression involving powers, such as \( 100^2 \), it's beneficial to confirm your calculation with a calculator. This process, called calculator verification, ensures that no simple arithmetic errors were made. Here’s how you can do it:

• Begin by entering the base into the calculator, which is 100.
• Utilize the exponent function, often represented by a button like \(^\wedge\) or \( x^y \) on the calculator.
• Enter the exponent, which is 2 in this instance.
• Complete the input, and the calculator will display the result as 10,000.

Calculator verification not only reassures you of your solution's correctness but also builds confidence in handling similar problems with larger numbers or different exponents in the future.