Problem 57 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. (Lesson 8.1) $$r^{2} \cdot r^{4}$$

Step-by-Step Solution

Answer

The expression \(r^{2} \cdot r^{4}\) as a single power of its base is \(r^{6}\).

Step 1: Identify the Powers

The expression has two powers of the base r: \(r^{2}\) and \(r^{4}\). Both powers have the same base which is r.

Step 2: Apply Exponent Addition Rule

According to the properties of exponents, when two powers with the same base are multiplied, the exponents are added together. So, \(r^{2} \cdot r^{4}\) can be rewritten as \(r^{2+4}\).

Step 3: Simplify the Exponent

Simplify the exponent by adding the numbers: \(2+4=6\). Thus, \(r^{2} \cdot r^{4}\) is equivalent to \(r^{6}\).

Key Concepts

Exponent Rules

Exponents are powerful mathematical tools that help us deal with repetitive multiplication efficiently. When working with exponents, there are a few important rules to keep in mind. One of the most fundamental is the "Product of Powers Rule." This rule states that when you multiply two powers with the same base, you simply add their exponents. For example, in the expression \( r^{2} \cdot r^{4} \), since both terms have the base \( r \), you add the exponents 2 and 4 to get the new exponent. Therefore, \( r^{2} \cdot r^{4} = r^{2+4} = r^{6} \).

• **Product of Powers Rule:** When multiplying two powers with the same base, add the exponents.
• This rule simplifies calculations by reducing complex multiplications to simple additions of exponents.
• Always ensure the bases are identical before applying this rule.

Understanding and applying these exponent rules is crucial for simplifying expressions and solving problems efficiently.

Multiplying Powers

Multiplying powers involves using specific rules to transform and simplify expressions. When you encounter multiplications like \( r^{2} \cdot r^{4} \), first identify the common base. Here, the base is \( r \). Since the bases match, you can use the exponent rule and add together the exponents.
**Why is this helpful?** Multiplying powers helps in:

• Reducing complex expressions into simpler forms.
• Making calculations easier and faster once you apply the exponent rules.
• Ensuring consistency in your working by following established mathematical rules.

By understanding how to multiply powers, you streamline your calculations, making them more straightforward and less error-prone. It's all about recognizing patterns and applying rules to efficiently manage multiple multiplications.

Simplifying Expressions

Simplifying expressions is all about taking complex mathematical problems and making them manageable. With exponents, you often face expressions that look intimidating at first glance. However, using the rules we've discussed, like the Product of Powers Rule, these expressions can be simplified.
In our example, \( r^{2} \cdot r^{4} \), instead of multiplying \( r \) multiple times, the process of simplifying allowed us to rewrite the expression into \( r^{6} \).

• **Simplification Process:** Identify common bases and use exponent rules to combine the exponents efficiently.
• Simplification leads to concise expressions that are easier to work with in further calculations or algebraic manipulations.
• Always double-check your work to ensure the simplification is accurate.

Simplifying expressions is a skill that improves with practice, leading to more efficient problem-solving and a deeper understanding of mathematics.