Question

Which number is the base in the expression \(8 \cdot 3^6 ?\) A. 3 B. 6 C. 8 D. \(3^6\)

Short answer

A. 3

Step-by-step solution

Understand the Problem

Determine which number acts as the base in the given expression. The base in an exponential expression is the number that is raised to a power.

Identify the Exponential Expression

Look at the term in the expression that involves an exponent. The term is \(3^6\) .

Define the Base

In the term \(3^6\) , the base is the number that is being raised to the power of 6. Therefore, the base is 3.

Verify the Answer

Check if option A, 3, matches our identified base for the exponent term \(3^6\). Yes, it does match.

Key concepts

Base

In an exponential expression, the base is the number that gets multiplied by itself. For example, in the expression \( 3^6 \), the base is the number 3. This means that 3 is to be multiplied by itself a number of times determined by the exponent. The base is crucial because it is the fundamental value that powers operate on.
When identifying the base, look for the number directly associated with the exponent. In the expression \( 8 \cdot 3^6 \), only the \( 3 \) has the exponent (6), making 3 the base. The number 8 is a coefficient in this context, not the base.

Exponent

The exponent in an exponential expression indicates how many times the base is to be multiplied by itself. It is written as a smaller number to the top right of the base. For instance, in the expression \( 3^6 \), the exponent is 6. This tells us that the base (3) should be used in a multiplication process 6 times: \( 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \).
Exponents can greatly change the value of an expression. Even a small base raised to a high exponent can yield very large numbers.

Power

The power in an exponential expression refers to the entire expression, encompassing both the base and the exponent. It represents the quantity obtained when the base is raised to the power indicated by the exponent. Taking our example, \( 3^6 \) is read as '3 to the power of 6'. The power gives us the full picture of the multiplication process dictated by the base and exponent.
To compute the power \( 3^6 \), you multiply the base (3) by itself as many times as indicated by the exponent (6). In this way, \( 3^6 = 729 \). Understanding powers helps simplify repeated multiplication and makes it easier to handle large calculations.