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Chapter 0: Problem 19

In Exercises \(1-28\), compute the numbers. $$ (-5)^{-1} $$

Short Answer

Expert verified
The answer to \((-5)^{-1}\) is \(\frac{1}{-5}\).

Step by step solution

01

Understand the problem

We need to compute \((-5)^{-1}\). This means raising -5 to the power of -1.
02

Apply the negative exponent rule

Recall that a negative exponent indicates taking the reciprocal of the base. So, \((-5)^{-1} = \frac{1}{-5}\).
03

Simplify the expression

The reciprocal of -5 is \(\frac{1}{-5}\). Therefore, \((-5)^{-1} = \frac{1}{-5}\).

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Key Concepts

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

Exponentiation
Exponentiation involves raising a number, called the base, to a particular power, which is known as the exponent. For example, in the expression \(a^b\), \(a\) is the base and \(b\) is the exponent.
The process can be broken down as follows:
  • If \(b\) is positive, you multiply the base \(a\) by itself \(b\) times.
  • If \(b\) is negative, you take the reciprocal of the base \(a\) and then raise it to the positive power \(-b\).
In the problem \((-5)^{-1}\), our base is -5, and our exponent \(-1\) tells us to find the reciprocal of -5.
Reciprocal
The reciprocal of a number is defined as 1 divided by that number. Essentially, it's flipping the numerator and the denominator of a fraction. For example, the reciprocal of 5 is \frac{1}{5}\.
Here's how to find the reciprocal:
  • For a whole number like 5, its reciprocal is \frac{1}{5}\.
  • For a fraction like \frac{3}{4}\, its reciprocal is \frac{4}{3}\.
In the case of \(-5\), its reciprocal is \frac{1}{-5}\.
Simplification
Simplification involves reducing a mathematical expression to its simplest form. It makes equations easier to understand and work with.
In this exercise, we start with \((-5)^{-1}\), apply the rule for negative exponents to find the reciprocal, and then simplify the expression.
Follow these steps:
  • Start with the expression \((-5)^{-1}\).
  • Recognize that the negative exponent means we need to take the reciprocal.
  • Therefore, \((-5)^{-1} = \frac{1}{-5}\).
As final simplification, our result stays \frac{1}{-5}\.

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Most popular questions from this chapter

Evaluate \(f(4)\). \(f(x)=x^{-1 / 2}\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\left(16 x^{8}\right)^{-3 / 4}\)

Compute the numbers. \(9^{1.5}\)

Convert the numbers from graphing calculator form to standard form (that is, without E). \(1.35 \mathrm{E} 13\)

In Exercises 110-113, convert the numbers from graphing calculator form to standard form (that is, without E). \(5 \mathrm{E}-5\)
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