Problem 8 - Algebra 1: Concepts and Skills

Question

Match the power with the words that describe it. $$ 6^{4} $$ A. four to the sixth power B. three to the seventh power C. seven to the third power D. six to the fourth power

Step-by-Step Solution

Answer

The short answer is D. six to the fourth power.

Step 1: Identify the base and the exponent

In the expression \(6^{4}\), the base is 6 and the exponent is 4.

Step 2: Look for the correct match

Now, from the provided options, we need to find the option that correctly describes the base and the exponent. The base is 6 (not 3, 4 or 7), and the exponent is 4 (not 6, 3 or 7).

Key Concepts

Base

In mathematics, when we talk about exponents, the concept of a "base" is fundamental. In an expression like \(6^4\), the base refers to the number that is being multiplied by itself. In this case, the base is 6. Understanding the base is important because it tells us which number is being repeatedly multiplied.

The base acts as the foundation of an exponential expression.

• It determines the repetitive multiplication that takes place.
• Serves as the main number you start with.

The base is essential when you need to evaluate or simplify powers because it identifies the number involved in the operation. Recognizing the base helps in understanding how the original quantity (here, 6) forms the bulk of the expression and shows which number is the subject of repetition as dictated by the exponent.

Exponent

In exponential expressions, the "exponent" plays a vital role and appears as the small number written above and to the right of the base, such as the 4 in \(6^4\). The exponent tells you how many times to multiply the base by itself.

Here are key elements about exponents:

• An exponent indicates repeated multiplication.
• It provides direction on how many times to use the base in a multiplication.

So, in \(6^4\), the exponent of 4 dictates that we should multiply 6 by itself four times: \[ 6 \times 6 \times 6 \times 6. \]
In understanding exponents, we essentially grasp the concept of scaling a number by itself multiple times, which can dramatically increase the value of the base number, depending on the size of the exponent.

Power

"Power" in mathematics is a term that describes the entire exponential expression, including both the base and the exponent. For example, in \(6^4\), the power is the full expression, representing the concept of raising a number (the base) to a specified exponent.
Understanding power involves grasping both the base and the exponent because together, they define the exponential operation.The term 'power' often confuses learners as it can refer to:

• The entire expression, like \(6^4\).
• The result of performing the exponential operation, which is 1296 for \(6^4\).

When discussing powers, such as 'six to the fourth power', we combine the ideas of both the base (6) and the exponent (4) to describe exponential expressions as a single concept. This conveys the idea of exponential growth or diminutive shrinking through repeated multiplication or division.