Problem 24 - Elementary and Intermediate Algebra

Question

Express using positive exponents and simplify, if possible. \(5^{-1}\)

Step-by-Step Solution

Answer

The expression is \(\frac{1}{5}\).

Step 1: Understanding the Problem

The task requires expressing the expression \(5^{-1}\) using positive exponents. A negative exponent indicates that the base should be reciprocated.

Step 2: Reciprocal of the Negative Exponent

To express \(5^{-1}\) as a positive exponent, recall that \(a^{-n} = \frac{1}{a^n}\). Therefore, \(5^{-1} = \frac{1}{5^1}\).

Step 3: Simplification

Since \(5^1\) simply equals 5, \(\frac{1}{5^1}\) simplifies to \(\frac{1}{5}\). The expression with a positive exponent is \(\frac{1}{5}\).

Key Concepts

Negative Exponents

Negative exponents can often seem confusing initially. However, they are simply a notation that signifies the inverse or reciprocal of a number raised to a positive power. When you see a negative exponent, such as in the expression \(5^{-1}\), it means that we are dealing with the reciprocal of the base raised to the positive of that exponent. The key idea is this: any number raised to a negative exponent can be rewritten as \(a^{-n} = \frac{1}{a^n}\).

To illustrate with \(5^{-1}\), instead of thinking of it as a complicated operation, switch the perspective to reciprocation. Essentially, it's about flipping the fraction.

• Think of \(5^{-1}\) as the inverse of \(5\).
• In equational terms: \(5^{-1}\) becomes \(\frac{1}{5^1}\).

This approach leads directly to the introduction and understanding of the reciprocal.

Reciprocal

The reciprocal is a pivotal concept in transforming negative exponents into readable and simpler expressions with positive exponents. But what exactly does 'reciprocal' mean? To find the reciprocal of a number, you essentially take 1 and divide it by that number. This is how we shift from a negative to a positive exponent.

Considering \(5^{-1}\), the reciprocal involves changing the base 5 into its reciprocal, giving us \(\frac{1}{5}\). In general:

• The reciprocal of \(a\) is \( \frac{1}{a} \).
• For \(a^{-n}\), the reciprocal isn’t just \(\frac{1}{a}\) but \(\frac{1}{a^n}\), reflecting the positive power.

This understanding aids in simplifying expressions and making them more digestible, especially when higher-level problems demand swift conversions from negative to positive exponents.

Simplification

Simplifying expressions with negative exponents is a crucial final step to ensure clarity and comprehensibility in mathematics. Once an expression has been converted from a negative to a positive exponent, it's about reducing it to its simplest form.

After rewriting \(5^{-1}\) as \(\frac{1}{5^1}\), we simplify by recognizing that \(5^1 = 5\). There's no further reduction required here because \(\frac{1}{5}\) is already in its simplest form. Some tips for simplification include:

• Check if the base raised to a positive power can be further reduced.
• Simplification might involve merely recognizing the base number, like in our example.

This simplification not only confirms accuracy but also aids in solving more complex equations with confidence. Keeping expressions simplified helps maintain orderliness and ensures smooth problem-solving processes.