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Chapter 5: Problem 34

Express using negative exponents. $$\frac{1}{2}$$

Short Answer

Expert verified
2^{-1}

Step by step solution

01

Understanding Negative Exponents

A negative exponent indicates that the base should be moved to the denominator and the exponent changed to positive. For example, if you have a base with a negative exponent, you can rewrite it as the reciprocal with a positive exponent.
02

Rewrite the Expression

The given expression is \(\frac{1}{2}\). Recognize that this can be rewritten using a negative exponent.
03

Apply the Negative Exponent Rule

According to the negative exponent rule, \(\frac{1}{2}\) can be rewritten as \(2^{-1}\). Here, 2 is the base and -1 is the exponent.

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Key Concepts

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

Reciprocal
The concept of the reciprocal is fundamental when dealing with negative exponents. A reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 2 is \[ \frac{1}{2} \].
When we talk about negative exponents, we're actually being directed to take the reciprocal of the base. For example, \[a^{-n} = \frac{1}{a^n} \]. So, if we have \[2^{-1} \], it means we take the reciprocal of 2, which is \[\frac{1}{2} \]. This rule applies to any base raised to a negative exponent.
Positive Exponent
A positive exponent indicates how many times you multiply the base by itself. For instance, \[2^3 \] means \[2 \times 2 \times 2 = 8 \]. Positive exponents are straightforward because they tell us to multiply the base repeatedly based on the exponent's value.
When we convert a negative exponent to its reciprocal, like \[2^{-1} \], it changes into a positive exponent in the denominator: \[2^{-1} = \frac{1}{2^1} \]. This helps in simplifying expressions and understanding the relation between negative and positive exponents.
Base
In any exponential expression, the base is the number that is being multiplied. For example, in \[2^3 \], 2 is the base, and it gets multiplied three times because of the exponent 3.
When dealing with negative exponents, understanding the base is essential because it helps in finding the reciprocal and rewriting the expression. Knowing the base allows us to apply the negative exponent rule accurately, like transforming \[2^{-1} \] into \[\frac{1}{2} \]. This process becomes simpler once we identify the base correctly. The base remains the same whether the exponent is positive or negative.

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Most popular questions from this chapter

Simplify. \(\left(3 x^{2}\right)^{3}+4 x^{2} \cdot 4 x^{4}-x^{4}(2 x)^{2}+\left((2 x)^{2}\right)^{3}-\) \(100 x^{2}\left(x^{2}\right)^{2}\)

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{a^{4}}{b^{3}}\right)^{5} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -16+5 $$

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For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{2}{x}\\\ &g(x)=x^{2}-4 \end{aligned} $$
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