Problem 23 - Elementary and Intermediate Algebra

Question

Express using positive exponents and simplify, if possible. \(6^{-1}\)

Step-by-Step Solution

Answer

\(6^{-1} = \frac{1}{6}\).

Step 1: Understand Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. So, the expression \(6^{-1}\) can be rewritten using its reciprocal.

Step 2: Rewrite Using Reciprocals

To express \(6^{-1}\) with a positive exponent, take the reciprocal of 6 raised to a positive exponent: \(6^{-1} = \frac{1}{6^1}\).

Step 3: Simplify the Expression

Since \(6^1 = 6\), the expression simplifies to \(\frac{1}{6}\). Thus, \(6^{-1}\) expressed with a positive exponent is \(\frac{1}{6}\).

Key Concepts

Understanding Reciprocals

Reciprocals are fundamental when it comes to dealing with negative exponents. Let's break it down. A reciprocal is essentially one of a pair of numbers that, when multiplied together, yield the number 1. To find the reciprocal of any number "a", you simply take 1 divided by "a", written as \( \frac{1}{a} \).
Now, when you see an expression like \( 6^{-1} \), the negative exponent tells you that the base—here 6—needs to turn into its reciprocal. It becomes \( \frac{1}{6} \).

• The reciprocal of 6 is \( \frac{1}{6} \).
• The reciprocal effectively inverts the number, meaning it swaps the position of the numerator and denominator if written as a fraction.

This process allows us to transform expressions with negative exponents into a form that is easier to manage.

Simplifying Expressions with Reciprocals

Simplification is all about making an expression as straightforward as possible. After using the reciprocal to turn \( 6^{-1} \) into \( \frac{1}{6} \), we already have a much simpler form. Simplifying expressions often requires:

• Understanding the operation: here, it was about using the reciprocal.
• Rewriting complex parts into easily manageable parts, such as reducing fractions if possible or recognizing when you've achieved a simplified expression.

Since \( 6^1 = 6 \), there are no further steps needed to simplify \( \frac{1}{6} \). You ended up with a clean, neat expression that represents the original question, but now with only positive exponents.

The Power of Positive Exponents

Positive exponents are often easier to handle than negative ones. They indicate how many times you need to multiply the base by itself. For example, with \( 6^1 \), you just have 6, because you're only multiplying 6 by itself once—not at all, really.
Switching from negative to positive exponents is about clarity, accuracy, and ease of calculation. Once transformed, expressions provide a clear path to calculation and understanding:

• Positive exponents hold a direct meaning of repeated multiplication.
• They ensure there's no fractional ambiguity, as fractions can remain intact when needed.
• In expressions with variables, having positive exponents means a better view of growth patterns.

Thus, rewriting negative exponents with positive ones aids in demystifying math, allowing for smoother operations and insights.