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Chapter 0: Problem 14

Compute the numbers. $$\left(\frac{1}{2}\right)^{-1}$$

Short Answer

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Step by step solution

01

Understand Negative Exponents

A negative exponent means to take the reciprocal of the base and then apply the positive exponent. The expression \(\bigg(\frac{1}{2}\bigg)^{-1}\) can be rewritten by applying this rule.
02

Apply the Reciprocal Rule

Taking the reciprocal of \(\frac{1}{2}\) gives \(2\). Therefore, \(\bigg(\frac{1}{2}\bigg)^{-1} = 2\).

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Key Concepts

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

Reciprocal
A reciprocal is what we get when we flip a number over. For instance, the reciprocal of \(\frac{1}{2}\) is \(2 \). When we have a fraction like \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). It's like turning it upside down!

In the expression \(\bigg(\frac{1}{2}\bigg)^{-1}\), understanding the reciprocal is the key. First, we identify the base \(\frac{1}{2}\), and then flip it to get its reciprocal: \(2\).

This is a useful concept because negative exponents mean we work with reciprocals. This helps in quickly turning complicated fractions into simpler numbers.
Base
The base is the number that is being multiplied by itself. In our case, from the exercise \(\bigg(\frac{1}{2}\bigg)^{-1}\), the base is \(\frac{1}{2}\).

The base is important because it tells us which number's reciprocal we need to find when dealing with negative exponents. Think of the base as the starting point. If you know your base, it's easier to apply rules like finding the reciprocal or raising it to any power.

For example, in \(4^3\), \4\ is the base, telling us that \4\ is multiplied by itself three times: \4 \times 4 \times 4.\ For negative exponents, recognizing the base helps in understanding what to flip when adapting to positive exponents.
Positive Exponent
When we have a positive exponent, it shows how many times we multiply the base by itself. For example, in \(5^2\), the positive exponent \(2\) means we multiply \5 \times 5\.

Positive exponents are straightforward compared to negative ones, as there is no need to find reciprocals. However, once we turn a negative exponent into a positive one by flipping the base, it behaves just like a positive exponent.

In our exercise, when \(\bigg(\frac{1}{2}\bigg)^{-1} = 2\), we've used the reciprocal rule to turn the negative \(-1\) exponent into a positive form. The positive exponent then means multiplying the new base \2\ just once, keeping it as \2\.

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Most popular questions from this chapter

Find the zeros of the function. (Use the specified viewing window.) $$f(x)=\frac{x}{x+2}-x^{2}+1 ;[-1.5,2] \text { by }[-2,3]$$

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