Problem 2 - Algebra and Trigonometry

Question

In the expression \(3^{4},\) the number 3 is called the _____, and the number 4 is called the _____.

Step-by-Step Solution

Answer

3 is the base, 4 is the exponent.

Step 1: Understanding Exponentiation Terms

In the expression equation \( a^b \), there are two main components: the base and the exponent. Our goal is to identify which term corresponds to each number in \( 3^4 \).

Step 2: Identify the Base

The base is the number that is being multiplied by itself. In the expression \( 3^4 \), the base is the number 3, as it is the number that will be repeatedly multiplied.

Step 3: Identify the Exponent

The exponent indicates how many times the base is multiplied by itself. In the expression \( 3^4 \), the number 4 is the exponent, which means 3 is multiplied by itself a total of 4 times.

Key Concepts

Base

In mathematics, the term **base** refers to the number that acts as the fundamental building block in an expression involving exponentiation. In every expression of the form \( a^b \), the base \(a\) is the number which will be multiplied by itself a certain number of times, determined by the exponent. You'll often hear the term "base" used when discussing power in mathematics.

Consider the expression \( 3^4 \). Here, the base is 3. This means the number 3 is what we're going to multiply by itself. As a helpful tip, always find the number beneath the exponent; that's your base! It's your starting point in understanding what value is being repeatedly used, which leads us nicely into discussing exponents.

Exponent

The **exponent** in a mathematical expression is a small, raised number that tells you how many times the base is multiplied by itself. In an expression like \( a^b \), the \(b\) is the exponent. It governs how many iterations of multiplication occur.

In our example \( 3^4 \), the exponent is 4. This indicates 3 is used in a multiplication process four times, like this: \(3 \times 3 \times 3 \times 3\).

• **Counting technique:** You can count the number of times you see the base written in the multiplication to verify you have used the exponent correctly.
• This number is like a multiplier for the multiplication process; the higher it is, the larger the final product, because you're multiplying more times.

Understanding exponents is critical because they help simplify long repeated multiplication into a concise expression.

Power in Mathematics

**Power** in mathematics is a term used to describe the expression of repeated multiplication within exponentiation. It combines both the base and the exponent. The power itself is what's calculated when the base is multiplied by the number of times specified by the exponent.

In the example \( 3^4 \), you calculate the power by performing the multiplication: \(3 \times 3 \times 3 \times 3 = 81\). Therefore, the power of 3 raised to the 4th is 81.

• The concept of power simplifies large numbers and extensive calculations.
• It's useful in many fields, from simple arithmetic to complex scientific calculations.

Power in mathematics takes exponentiation from theoretical to practical as you solve real problems by finding values resulting from these operations.