Problem 31 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(4^{2}\)

Step-by-Step Solution

Answer

The value of \(4^2\) is 16.

Step 1: Understand the Power Notation

The expression \(4^2\) consists of a base number (4) and an exponent (2). The exponent indicates how many times the base is multiplied by itself.

Step 2: Multiply the Base by Itself

To calculate \(4^2\), multiply the base number (4) by itself the number of times indicated by the exponent (2). So, compute \(4 \times 4\).

Step 3: Perform the Multiplication

Calculate \(4 \times 4\). The result of this multiplication is 16.

Step 4: Verify Using a Calculator

Use a calculator to verify the result. Enter the expression as 4 raised to the power of 2 (often entered as 4^2 or using a power key), and check that the result is indeed 16.

Key Concepts

Power Notation

In mathematics, power notation is a shorthand way to show that a number, known as the base number, is multiplied by itself a certain number of times. It consists of two components: a base and an exponent.

• The **base number** is the value that gets multiplied.
• The **exponent** is a small number written above and to the right of the base, showing how many times to use the base as a factor.

For example, in the expression \(4^2\), "4" is the base number, and "2" is the exponent. This notation indicates that 4 is used as a factor two times, which we will see how to calculate below.

Base Number

The base number is a key part of power notation and plays a vital role in understanding how powers work. It is the number we multiply by itself when we are using exponents.
Imagine the base number as the starting block of construction—it forms the foundation of the power expression.
In \(4^2\), 4 is the base number. Here's how you work with it:

• Identify the base: In \(4^2\), our base is 4.
• Understand its role: The base will be multiplied by itself as many times as the exponent indicates. Here, since our exponent is 2, we will multiply 4 by itself once.

The concept of the base number is fundamental to performing calculations with exponents. It determines the repeating factor in the expression.

Exponentiation

Exponentiation is the process of multiplying a base number by itself for a number of times determined by the exponent.
This concept is widely used not only in basic arithmetic but also in algebra and more advanced areas of mathematics.
To understand exponentiation better, consider the function it serves:

• **Simplification**: Rather than writing down multiple multiplications, exponentiation provides a concise way of expressing repeated multiplication.
• **Computation**: To compute \(4^2\), take the base number 4 and multiply it by itself according to the exponent. Thus, \(4 \times 4 = 16\).

In essence, exponentiation provides efficiency and simplicity in mathematical expressions.

Multiplication Verification

Verification through multiplication is an important step to ensure that the exponentiation is computed correctly.
After performing calculations manually, using a calculator to verify your answer helps confirm your understanding.
Here's how:

• **Manual Calculation**: After computing \(4 \times 4\) manually to get 16, you've completed the first step.
• **Using a Calculator**: Enter the power expression (\(4^2\)) in a calculator. This can be done by finding the power function, or directly inputting the expression if your calculator allows it. The calculator should yield 16.

This step ensures accuracy and boosts confidence, especially when handling more complex expressions.