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Chapter 1: Problem 3

Evaluate each expression without using a calculator. $$ 2^{-4} $$

Short Answer

Expert verified
The value of \(2^{-4}\) is \(\frac{1}{16}\).

Step by step solution

01

Understanding Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal raised to the positive of that exponent. Therefore, the expression \(2^{-4}\) can be rewritten as \(\frac{1}{2^4}\).
02

Evaluating the Positive Exponent

Now, calculate \(2^4\). Exponentiation means multiplying the base by itself the number of times indicated by the exponent. Hence, \(2^4 = 2 \times 2 \times 2 \times 2 = 16\).
03

Using Reciprocal to Solve the Expression

Since \(2^{-4} = \frac{1}{2^4}\), substitute \(2^4\) with \(16\). Thus, \(2^{-4} = \frac{1}{16}\).

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Key Concepts

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

Understanding Exponentiation
Exponentiation is a fundamental mathematical operation that involves raising a base number to the power of an exponent. In simple terms, it means multiplying the base by itself as many times as the number indicates on the exponent. For instance, when we talk about \(2^4\), we mean that you multiply 2 by itself four times: \(2 \times 2 \times 2 \times 2\). This results in 16.

Remember, the base is the number that appears at the bottom, while the exponent is written as a small number above and to the right of the base.
  • Base: the number being multiplied
  • Exponent: indicates how many times to multiply the base by itself
It's crucial to get comfortable with this operation, as it lays the foundation for more complex mathematical concepts and calculations.
The Concept of Reciprocal
A reciprocal is simply the inverse of a number. It flips the number from numerator to denominator and vice versa. If you have a fraction like \( \frac{a}{b} \), then the reciprocal will be \( \frac{b}{a} \). When it comes to whole numbers like 2, its reciprocal is \( \frac{1}{2} \).

In the context of negative exponents, the reciprocal helps us understand what to do. A negative exponent (-1) signals the need to take the reciprocal of the base. So, \(2^{-1}\) becomes \(\frac{1}{2^1}\). This flips the base while changing the exponent from negative to positive.

Understanding the reciprocal is key to simplifying and evaluating expressions that include negative exponents.
Evaluating Expressions with Negative Exponents
Evaluating expressions with negative exponents involves both understanding exponentiation and the reciprocal. When you encounter an expression such as \(2^{-4}\), the negative sign tells you to first find the reciprocal of the base. So, we transform \(2^{-4}\) into \(\frac{1}{2^4}\).

Next, calculate the positive exponent. As we did earlier, \(2^4 = 16\). Thus, continuing with our example, the expression becomes \(\frac{1}{16}\).

Here are the steps broken down for clarity:
  • Identify the negative exponent.
  • Calculate the reciprocal of the base and remove the negative sign from the exponent.
  • Evaluate the positive exponent to complete the expression.
This process allows you to accurately and confidently evaluate expressions with negative exponents, ensuring you understand each step involved.

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Most popular questions from this chapter

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The Apocryphal Manufacturing Company makes blivets out of widgets. If a linear function \(f(x)=m x+b\) gives the number of widgets that can be made from \(x\) blivets, what are the units of the slope \(m\) (widgets per blivet or blivets per widget)?

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