Problem 11 - Elementary Algebra

Question

For the following problems, expand each product (do not compute the actual value). \(3^{4}\)

Step-by-Step Solution

Answer

Question: Expand the expression \(3^4\) by writing it as a product of the same base being multiplied as many times as the exponent indicates. Answer: \(3^4 = 3 \times 3 \times 3 \times 3\)

Step 1: Rewrite the expression with the base multiplied by itself according to the exponent

The expression \(3^4\) means that we need to multiply the base (3) by itself for a total of 4 times: \[3^4 = 3 \times 3 \times 3 \times 3.\]

Step 2: Check your result

We have now expanded the expression \(3^4 = 3 \times 3 \times 3 \times 3\). Make sure that the base (3) is multiplied by itself 4 times as indicated by the exponent.

Key Concepts

Expansion

Expansion in the context of exponentiation involves rewriting a power expression by expressing the base as a product multiplied by itself for a certain number of times. This process is often used in algebra to simplify, compare, or manipulate expressions. For example, the expression \(3^4\) can be expanded to \(3 \times 3 \times 3 \times 3\).
This expanded form shows the repeated multiplication clearly, making it easier to understand the expression's structure. Expansion is also helpful when you need to apply additional operations to each factor or when visualizing how the product grows as the exponent increases.

Base

The base in an expression involving exponents is the number that is being raised to a power. In the expression \(3^4\), the number 3 is the base.
The base is the starting point of multiplication, determining how many times it should be multiplied by itself as indicated by the exponent.

• The choice of base can greatly affect the outcome of the expression when expanded.
• The same exponent applied to different bases will yield different results, showcasing the importance of understanding the base in calculations.

Generally, knowing the base is crucial because it is the foundation of interpreting the whole exponentiation operation.

Multiplication

Multiplication is a fundamental arithmetic operation used in the expansion of exponentiation expressions. When you have a power, such as \(3^4\), you are essentially performing the multiplication of the base number repeatedly.
This example involves multiplying the base, 3, four times: \(3 \times 3 \times 3 \times 3\).
In this process:

• Each multiplication step contributes to the growth of the product.
• Multiplication here is about swiftly combining numbers, which increases the size or magnitude of the result.

Understanding this operation is key to properly executing the steps in expanding power expressions.

Exponents

Exponents are a mathematical tool used to indicate how many times a base is to be multiplied by itself. They simplify the expression of large power products. Consider \(3^4\), where 4 is the exponent.
This means multiplying 3 by itself four times. Exponents are vital for several reasons:

• They provide a concise way to represent repeated multiplication without having to write the whole expansion.
• Exponents make it easier to perform and manage calculations, particularly with large bases or high powers.

In other words, exponents streamline math operations, making them more manageable and organized.