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Chapter 6: Problem 5

Evaluate using the rules of exponents. $$6^{-1}$$

Short Answer

Expert verified
The short answer based on the given solution is: Applying the negative exponent rule for the expression \(6^{-1}\), we obtain \(6^{-1} = \frac{1}{6^1} = \frac{1}{6}\).

Step by step solution

01

Identify the base and exponent

In our given expression, \(6^{-1}\), the base is \(6\), and the exponent is \(-1\).
02

Apply the negative exponent rule

Now, we will apply the negative exponent rule: \(a^{-n} = \frac{1}{a^n}\) In our expression, we have \(6^{-1}\), so our \(a = 6\) and \(n = 1\). Using the rule, we get: \(6^{-1} = \frac{1}{6^1}\)
03

Simplify the expression

Now, we will simplify the expression: \(\frac{1}{6^1} = \frac{1}{6}\) So, the final answer is \(\boldsymbol{\frac{1}{6}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluating Expressions
Evaluating expressions is the process of finding the value of an algebraic statement. To evaluate expressions with exponents, it's important to carefully follow any applicable rules.
In this exercise, you are asked to evaluate the expression \(6^{-1}\). The trick here is to understand how a negative exponent influences the calculation.
  • Identify the base and exponent. For instance, in \(6^{-1}\), the base is 6 and the exponent is -1.
  • Apply any relevant rules of exponents to simplify the expression. In this case, knowing how to handle negative exponents will help you find the solution.
This approach ensures that the expression is simplified down to its simplest form for an easy evaluation.
Rules of Exponents
The rules of exponents are mathematical guidelines that help in manipulating expressions with exponents easily. They are crucial when working with expressions like \(6^{-1}\).
  • Negative Exponents Rule: This rule explains that any base \(a\) raised to a negative exponent \(-n\) can be rewritten as \(\frac{1}{a^n}\). For instance, \(6^{-1}\) can be changed to \(\frac{1}{6^1}\).
  • Zero Exponent Rule: Any base raised to the zero power is 1, i.e., \(a^0 = 1\).
  • Product of Powers Rule: When multiplying two exponents with the same base, you add the exponents, like \(a^m \times a^n = a^{m+n}\).
For this problem, focusing on the Negative Exponents Rule is key, as it directly simplifies \(6^{-1}\) into a more manageable form.
Exponent Properties
Exponent properties provide a framework for understanding how exponential expressions behave mathematically. They鈥檙e essential for solving equations effectively.
Here are some useful properties to remember:
  • Positive vs. Negative Exponents: A positive exponent denotes multiplication, while a negative exponent indicates division or reciprocal. For instance, \(6^1\) means 6, whereas \(6^{-1}\) means \(\frac{1}{6}\).
  • Multiplicative Inverse: Negative exponents are tied to the concept of the multiplicative inverse, which is the value you multiply a number by to get 1. For instance, the multiplicative inverse of 6 is \(\frac{1}{6}\), aligning with \(6^{-1} = \frac{1}{6}\).
Understanding these properties allows you to confidently manipulate and simplify expressions involving exponents, such as \(6^{-1}\), resulting in easy evaluation.

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