Problem 7 - Algebra 1

Question

Use the product of powers property to simplify the expression. $$ 3^{2} \cdot 3^{5} $$

Step-by-Step Solution

Answer

The simplification of \(3^{2} \cdot 3^{5}\) is \(3^{7}\).

Step 1: Identify the base and the exponents

The expression presented is \(3^{2} \cdot 3^{5}\). Here, the base of the power is 3 in both \(3^{2}\) and \(3^{5}\). The exponents for these powers are 2 and 5 respectively.

Step 2: Apply the product of powers property

The product of powers property states that \(a^{m} \cdot a^{n} = a^{m+n}\), where a is the common base and m and n are the exponents. This property states that when multiplying two powers with the same base, you can add the exponents. In this case, \(3^{2} \cdot 3^{5}\) becomes \(3^{2+5}\).

Step 3: Calculate the new exponent

Add the exponents 2 and 5 to find the exponent in the simplified expression. Doing so gives us \(3^{2+5} = 3^{7}\).

Key Concepts

Exponent Rules

Understanding Exponent Rules is a fundamental aspect of working with powers in algebra. They provide a framework for simplifying expressions involving exponents, ensuring that calculations are both accurate and efficient. One key rule is the Product of Powers Property. This states that when you multiply terms with the same base, you can simply add their exponents. For example, in the expression \(3^2 \cdot 3^5\), both terms have the base 3. Applying the property means combining the exponents: \(2 + 5\), resulting in \(3^7\).The Product of Powers Property can be summarized as:

• If the bases are the same: \(a^m \cdot a^n = a^{m+n}\).
• This property simplifies multiplying powers swiftly without dealing with each term separately.

Knowing these rules helps you to handle complex expressions seamlessly, and is particularly useful in algebra and higher mathematics.

Simplifying Expressions

Simplifying Expressions is a crucial process that involves rewriting them in their simplest or most efficient forms. The goal is often to reduce complexity while maintaining equivalence.In the context of the given exercise, you start with \(3^2 \cdot 3^5\). By identifying common bases and applying the Product of Powers Property, the expression is simplified to \(3^{7}\). This process eliminates unnecessary complexity, making it easier to understand and work with.Key steps in simplifying:

• Identify common elements within the expression, like common bases for powers.
• Apply relevant algebraic properties to reduce the expression.
• Simplify step-by-step to maintain accuracy.

This approach makes algebraic manipulations more approachable and reduces error potential.

Algebraic Properties

Algebraic Properties are foundational rules that govern the operations on algebraic expressions and equations. They provide consistency and structure, allowing us to predictably manipulate mathematical expressions.In simplifying \(3^2 \cdot 3^5\), the Product of Powers is one of these properties. It exemplifies the broader category of exponents properties that include:

• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \), for subtracting exponents.
• Power of a Power: \((a^m)^n = a^{mn}\), for multiplying exponents over another power.

These properties help in maintaining order and simplifying problems that would otherwise be cumbersome. Understanding these can ease the learning curve in advanced topics like calculus, where manipulation of expressions is frequent.