Problem 47 - Algebra A Combined Function

Question

Evaluate. $$ 4^{2} $$

Step-by-Step Solution

Answer

16

Step 1: Identify the Base and Exponent

In the expression \(4^2\), the number 4 is the base, and the number 2 is the exponent. The exponent tells us how many times we need to multiply the base by itself.

Step 2: Interpret the Expression

The expression \(4^2\) means we need to multiply 4 by itself 2 times. This can be rewritten as \(4 \times 4\).

Step 3: Perform the Calculation

Multiply the base by itself: \(4 \times 4 = 16\). This is the value of \(4^2\).

Key Concepts

Base and Exponent

In mathematics, when we talk about exponentiation, we often use the terms \(\text{base}\) and \(\text{exponent}\). The base is the number that is being multiplied, and the exponent indicates how many times the base is used in the multiplication. For example, in the expression \(4^2\), 4 is the base and 2 is the exponent. This effectively tells us to multiply 4 by itself, 2 times.

Understanding these roles is crucial:

• The base is the primary number we are working with. It is the number to be repeatedly multiplied.
• The exponent signifies the number of times the base is utilized as a factor in multiplication.

It's like giving instructions; the base tells you what to work with, while the exponent tells you how many times to repeat it.

Multiplication

Multiplication is the arithmetic operation of scaling one number by another. In the context of exponentiation, it plays a significant role because the exponent signifies multiple instances of multiplication of the base. In simpler terms, when you see an expression like \(4^2\), it is telling you to multiply 4 by itself.

Let's break down how multiplication works here:

• First, recognize the base (4), which is the number being multiplied.
• Secondly, the exponent (2) indicates how many times 4 should appear in the multiplication — \(4 \times 4\).
• Finally, perform the multiplication: \(4 \times 4 = 16\).

This process is straightforward, but it's important to understand each step to master multiplication and apply it correctly.

Powers of Numbers

The concept of powers of numbers refers to using exponentiation to calculate expressions where a number is multiplied by itself a specified number of times. Commonly referred to as "raising a number to a power," this concept simplifies repeated multiplication into a concise notation.

When we say "the power of a number," we mean:

• The base number is multiplied by itself.
• The exponent signals how many times this multiplication occurs.
• For instance, \(4^2\) means 4 is multiplied by itself, resulting in 16.

Using powers simplifies math problems by reducing lengthy multiplications into compact formulas and makes calculations quicker and more efficient.