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Chapter 4: Problem 14

Write each expression using a positive exponent. $$4^{-1}$$

Short Answer

Expert verified
The expression with a positive exponent is \(\frac{1}{4}\).

Step by step solution

01

Identifying the Negative Exponent

The given expression is \(4^{-1}\). We need to understand that the negative exponent indicates the reciprocal of the base raised to the positive exponent.
02

Writing the Reciprocal Form

To convert \(4^{-1}\) into an expression with a positive exponent, we write it as \(\frac{1}{4^1}\). This is because \(a^{-n} = \frac{1}{a^n}\).
03

Simplifying the Expression

Since \(4^1 = 4\), the expression \(\frac{1}{4^1}\) simplifies to \(\frac{1}{4}\). This is the expression of \(4^{-1}\) with a positive exponent.

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Key Concepts

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

Understanding Reciprocals
The concept of a reciprocal is essential when working with negative exponents. In mathematics, the reciprocal of a number is like flipping the number over, similar to turning a fraction upside down. For instance, the reciprocal of a whole number like 4 is simply turning it into a fraction:
  • 4 becomes \( \frac{1}{4} \)
When we talk about reciprocals in the context of exponents, we're indicating how a negative exponent transforms an expression. If you see an expression like \(4^{-1}\), you apply the reciprocal rule:
  • The expression becomes \( \frac{1}{4^1} \), following the formula \(a^{-n} = \frac{1}{a^n}\)
This understanding helps simplify complex mathematical expressions, thereby making calculations easier to handle.
Grasping Positive Exponents
Exponents are a shorthand way to express repeated multiplication. A positive exponent indicates the number of times a base is multiplied by itself. For example, in our expression, converting a negative exponent into a positive exponent improves clarity and simplicity.
  • In \(4^{-1}\), by transforming the expression to \(\frac{1}{4^1}\), we observe this rule in practice.
  • You are left with a positive exponent, \(4^1\), which is much simpler to interpret because \(4^1 = 4\)
This transformation process aids in maintaining expressions concise and easy to calculate, leaving us no ambiguity in understanding what operations need to be performed.
Simplification of Expressions
Simplifying expressions is a key part of algebra that makes complex problems easier to solve. When dealing with exponents, simplifying entails expressing components using positive exponents and merging any possible terms together.
  • After identifying the negative exponent, \(4^{-1}\), we transformed it into its reciprocal form, \(\frac{1}{4^1}\).
  • Then, simplifying \(4^1\) directly gives us \(4\), leaving the entire expression as \(\frac{1}{4}\)
Each simplification step reduces complexity, paving the way to more manageable calculations. It鈥檚 all about breaking down expressions into their simplest forms, ensuring a clear path to problem resolution.

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