Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(3^{2} \times 3^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(3^5\).

Step 1: Understand the Question

We are asked to simplify the expression \(3^{2} \times 3^{3}\). This involves using the properties of exponents to combine these terms into one expression with a positive exponent.

Step 2: Apply the Product of Powers Property

The Product of Powers Property states that when you multiply two exponents with the same base, you can add their exponents. In this case, the base is 3 for both terms, so you can write \(3^{2} \times 3^{3} = 3^{2+3}\).

Step 3: Simplify the Expression

Add the exponents from the previous step: \(2 + 3 = 5\), so the expression becomes \(3^{5}\). This is the expression simplified with positive exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions involves transforming a mathematical expression into a simpler or more comprehensible form. It often means to perform all possible arithmetic operations and combine like terms to reduce the expression to its simplest form. When you simplify, you make it as concise as possible without changing its value. This process is crucial because it helps in understanding and solving mathematical problems efficiently.
In our given example, we are tasked to simplify the expression \(3^{2} \times 3^{3}\). To do this effectively, we will rely on exponent rules, which allow us to combine and reduce expressions involving powers. Recognizing the base and grouping terms appropriately allows an efficient simplification, making calculations easier and clearer.

Product of Powers Property

The `Product of Powers Property` is a fundamental rule in algebra involving exponents. It states that when you have the same base in multiplication, you add the exponents.
For the expression \(3^{2} \times 3^{3}\), both terms have the base 3. According to this property, you can combine them by adding their exponents: \(3^{2+3}\). Here’s how it works step by step:

• Identify the base: In this case, both are 3.
• Check that you're multiplying terms with the same base.
• Add the exponents together: \(2 + 3 = 5\).
• Write the result as: \(3^{5}\).

This simplification makes it clear and easy to understand, showing how efficiently we can work with powers.

Positive Exponents

An exponent indicates how many times you multiply a number, called the base, by itself. A positive exponent is simply a straightforward expression of this repetitive multiplication. For example, \(3^5\) means \(3 \times 3 \times 3 \times 3 \times 3\).
Ensuring exponents remain positive is important in simplifying expressions, as it allows for consistent interpretation and computation. The original exercise required us to express the final simplified expression \(3^5\) with positive exponents. This avoids the complexity and potential errors that might arise from negative or zero exponents.
When tackling mathematical problems involving exponents, always aim for solutions that express your answers with positive exponents unless specifically asked otherwise. This makes your answer simple, clear, and widely acceptable in mathematical communications.