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Chapter 15: Problem 16

Write an equivalent expression using negative exponents. \(\frac{1}{n}\)

Short Answer

Expert verified
n^{-1}

Step by step solution

01

Understand the expression

The given expression is a fraction \( \frac{1}{n} \). It involves the number 1 divided by some variable or number, denoted as \( n \).
02

Recall the definition of negative exponents

Negative exponents are defined such that \( a^{-m} = \frac{1}{a^m} \). This means that any number or variable raised to a negative exponent is equivalent to 1 divided by that number or variable raised to the positive equivalent of the exponent.
03

Apply the definition

Using the rule of negative exponents, express \( \frac{1}{n} \) using negative exponents. According to the definition, we have \( n^{-1} = \frac{1}{n} \).

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Key Concepts

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

Understanding Fractions
A fraction represents a part of a whole and is written in the form \(\frac{a}{b}\). In this notation, \(a\) is the numerator (the top part) and \(b\) is the denominator (the bottom part). For instance, the fraction \(\frac{1}{2}\) means you have one out of two equal parts.
The fraction \(\frac{1}{n}\) defines that you have one part out of \(n\) equal parts. Understanding fractions is essential when working with exponents, including negative exponents.
A fraction like \(\frac{1}{n}\) can also be seen as a division problem where 1 is divided by \(n\). If \(n=2\), \(\frac{1}{n}\) would equal 0.5.
What is a Variable?
In mathematics, a variable is a symbol (often a letter) that stands for a number which can vary or change. For example, in the expression \(\frac{1}{n}\), \(n\) is a variable. It can be any number.
Variables are essential because they let us write general rules or formulas. Instead of working with just specific numbers, we can work with a range of numbers.
When \(n\) changes, the value of the fraction \(\frac{1}{n}\) changes too. This concept is very powerful as it allows for the creation of flexible and general mathematical expressions.
Definition of Negative Exponents
Negative exponents might seem confusing at first, but they follow a simple rule: \(a^{-m} = \frac{1}{a^m}\). This means that raising a number to a negative exponent is the same as taking the reciprocal of that number raised to a positive exponent.

For example, \(2^{-3}\) is the same as \(\frac{1}{2^3}\), which simplifies to \(\frac{1}{8}\).
Here are some key points to remember:
  • A negative exponent flips the number or variable.
  • \(n^{-1}\) simply means \(\frac{1}{n}\).
This is how the fraction \(\frac{1}{n}\) can be rewritten using negative exponents:
\(\frac{1}{n} = n^{-1}\).

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