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Chapter 1: Problem 27

Simplify the given expressions. Express results with positive exponents only. $$6^{-1}$$

Short Answer

Expert verified
\(\frac{1}{6}\)

Step by step solution

01

Understand Negative Exponents

A negative exponent indicates taking the reciprocal of the base. For a general expression with a negative exponent, such as \(a^{-n}\), we can rewrite it as \(\frac{1}{a^n}\). This helps us convert negative exponents into positive ones.
02

Apply the Rule to the Expression

The given expression is \(6^{-1}\). Applying the rule for negative exponents, rewrite this as the reciprocal of \(6\) raised to the positive power of 1: \(\frac{1}{6^1}\).
03

Simplify the Expression

Simplify the expression from Step 2. Since \(6^1\) is just \(6\), the reciprocal is \(\frac{1}{6}\). This is already in its simplest form, with positive exponents.

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Key Concepts

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

Reciprocal Function
When you encounter a negative exponent, it signifies that you need to determine the reciprocal of the base number. Think of the reciprocal as flipping the number.
For instance, the reciprocal of a positive number like 5 is simply \( \frac{1}{5} \).
But how does this relate to exponents?
  • With exponents, if you see \( 6^{-1} \), the negative sign lets us know that we should "inverse" the 6, essentially flipping it.
  • This means converting \( 6^{-1} \) into its reciprocal, which is \( \frac{1}{6} \).
By understanding this, you can easily change negative exponents into positive ones by applying the reciprocal function rule. The change to positive exponents makes further calculations and comprehension much simpler.
Simplifying Expressions
Simplifying expressions is a crucial skill in algebra and mathematics as a whole. It involves rewriting expressions in a more straightforward or more recognizable form.
  • To simplify \( 6^{-1} \), we first change the negative exponent to positive by finding the reciprocal, resulting in \( \frac{1}{6} \).
  • Since \( 6^1 = 6 \), this simplification requires no further steps because \( \frac{1}{6} \) is already the simplest form.
Always check if an expression can be further simplified, though in this case, "6" is a prime number and doesn't break down further. Clear and concise expressions are not only simpler to read but also more efficient to use in calculations.
Exponent Rules
Understanding exponent rules is key to mastering algebra and solving mathematical problems efficiently. Exponent rules tell us how to handle powers, including both positive and negative exponents.
  • Negative exponents indicate that you should take the reciprocal of the base. For instance, \( a^{-n} \) becomes \( \frac{1}{a^n} \).
  • Remember that \( a^n \cdot a^{-n} \) simplifies to 1 since they are reciprocal operations.
  • Positive exponents demonstrate repeated multiplication, such as \( 2^3 = 2 \times 2 \times 2 \).
Often, the challenge is deciding which rule to apply. With practice, determining whether to reciprocate or simplify becomes second nature. Always ensure that final expressions use positive exponents unless instructed otherwise, enabling them to be intuitive and easy to interpret.

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