Problem 35 - College Algebra with Modeling and Visualization

Question

Exercises 35-44: Use the product rule to simplify. $$ 6^{3} \cdot 6^{-4} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{6}\).

Step 1: Understand the Powers of Base

The expression given is a multiplication of terms, both having the same base "6". Each number is represented as a power of 6: \(6^3\) and \(6^{-4}\).

Step 2: Apply the Product Rule for Exponents

The product rule for exponents states that when multiplying two exponents with the same base, you can add the exponents: \(a^m \cdot a^n = a^{m+n}\). Here, \(m = 3\) and \(n = -4\).

Step 3: Calculate the New Exponent

Add the exponents 3 and -4: \[3 + (-4) = -1\]

Step 4: Rewrite the Expression

Replace the original multiplication with a single power of 6: \[6^{3} imes 6^{-4} = 6^{-1}\]

Step 5: Simplify the Negative Exponent

The expression \(6^{-1}\) is equivalent to \(\frac{1}{6}\) because a negative exponent indicates the reciprocal: \[6^{-1} = \frac{1}{6}\]

Key Concepts

Powers of Base

When dealing with expressions involving exponents, it is crucial to comprehend the concept of a common base. A base is the number that is repeatedly multiplied by itself. In our example, the base is 6. We have two terms, both expressed with the base 6: \(6^3\) and \(6^{-4}\). This means:

• \(6^3\) is the base 6 raised to the power of 3, which equals \(6 \times 6 \times 6\).
• \(6^{-4}\) is the same base (6), raised to a different power, -4.

Understanding that both expressions have the same base allows us to apply the product rule, simplifying our calculations. Recognizing the powers of a common base is a foundational step in managing expressions with exponents.

Negative Exponents

Negative exponents might initially seem puzzling, but they follow a simple rule related to the reciprocal of the base. A negative exponent indicates that we should take the reciprocal of the base raised to the positive of that exponent. Here is how it works:

• An expression like \(6^{-1}\) is the same as \(\frac{1}{6^1}\), which simplifies to \(\frac{1}{6}\).
• More generally, for any non-zero base \(a\) and positive integer \(n\), \(a^{-n} = \frac{1}{a^n}\).

This interpretation of negative exponents is crucial for simplifying expressions, as it transforms expressions from products to simpler fractions, making further calculations easier.

Simplifying Expressions

Simplifying expressions involves making them easier to work with or understand. In the context of exponents, we apply rules like the product rule and the understanding of negative exponents to simplify.

• Using the product rule: When multiplying powers of the same base, add the exponents. For example, \(6^3 \cdot 6^{-4}\) becomes \(6^{3+(-4)} = 6^{-1}\).
• Handling negative exponents: Rewrite any negative exponent as a reciprocal. This means \(6^{-1} = \frac{1}{6}\).

By applying these steps, you're left with a simpler expression, \(\frac{1}{6}\), making calculations straightforward and more manageable. The key is to gradually apply these simplifying rules to express the original terms more concisely.