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Chapter 0: Problem 14

In Exercises \(1-28\), compute the numbers. $$ \left(\frac{1}{2}\right)^{-1} $$

Short Answer

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Step by step solution

01

Understand the Problem

Identify the given number and its mathematical operation. In this case, we're dealing with \(\frac{1}{2}\) and an exponent of -1.
02

Apply the Negative Exponent Rule

Recall that a negative exponent means taking the reciprocal of the base. So, \(\frac{1}{2}\right)^{-1}\) becomes \(\frac{2}{1}\).
03

Simplify the Expression

Simplify \(\frac{2}{1}\) to get 2.

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Key Concepts

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

Understanding Negative Exponents
Negative exponents can seem confusing at first, but they're actually straightforward once you get the hang of them. When you see a negative exponent, it means you need to take the reciprocal of the base. For instance, if you have \(\frac{1}{2}^{-1}\), it translates to \(\frac{1}{2}\) raised to the power of -1.
Instead of sticking with the negative exponent, you can simply flip the fraction. Therefore, \(\frac{1}{2}^{-1}\) really means \(\frac{2}{1}\), which simplifies to 2.
Breaking it down a bit more:
  • The negative exponent tells us to take the reciprocal of the fraction.
  • The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\).
  • Once the fraction is flipped, the exponent becomes positive, and you can then simplify the fraction if needed.
This method applies not just to fractions but to whole numbers and other expressions as well.
Exploring Reciprocal
Reciprocal is a key term you will encounter with negative exponents. A reciprocal of a number is simply flipping the numerator and the denominator. Think of it as the number you multiply by to get the product of 1.
For fractions, finding the reciprocal is easy:
  • Take \(\frac{a}{b}\).
  • Flip it to \(\frac{b}{a}\).
  • This new fraction, \(\frac{b}{a}\), is the reciprocal of \(\frac{a}{b}\).
For our example \(\frac{1}{2}\), the reciprocal would be \(\frac{2}{1}\), or just 2. This technique is helpful for many areas of math, including division of fractions, solving equations, and evaluating expressions with negative exponents. Remember: the reciprocal makes the negative exponent disappear, making it easier to work with.
Simplifying Fractions
Simplifying fractions is an essential skill in math. Once you've mastered it, many problems become easier.
Here鈥檚 a quick guide:
  • First step: Identify the numerator and the denominator. In our case, \(\frac{2}{1}\).
  • Second step: Check if you can divide both the numerator and the denominator by the same number鈥攖his is called finding the greatest common divisor (GCD). But here, since the fraction is already in its simplest form (numerator is 2 and denominator is 1), there's no further simplification needed.
  • The final step: Simplify to make the fraction as basic as possible. For \(\frac{2}{1}\), it simplifies directly to 2.
The general rule is: always look to simplify fractions when possible. It makes calculations and understanding the numbers much easier. For the problem at hand, \(\frac{2}{1} = 2\), so it's in its simplest form and easy to interpret.

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