Problem 13 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression to four decimal places using a calculator. $$e^{3}$$

Step-by-Step Solution

Answer

The value of \(e^3\) rounded to four decimal places is 20.0855

Step 1: Identify the Euler's number and power

In the given expression \(e^3\), \(e\) is the Euler's number, and the whole expression means \(e\) times itself twice (because the power is 3).

Step 2: Perform the operation on the calculator

Most scientific calculators will have a button for \(e\) to directly input it. After pressing the \(e\) button, press the exponent button, usually represented by '^', and then input 3. Press '=' or any button that equates the expression.

Step 3: Round to four decimal places

The calculator will give the answer, but it may have many decimal places. Because the problem asks for the answer to four decimal places, round your calculator's answer to that precision. This can be achieved by considering the digit at the fifth decimal place. If it is 5 or greater than 5, then add 1 to the digit at the fourth decimal place. If it is less than 5, leave the digit as it is.

Key Concepts

Euler's Number

Euler's number, denoted as \( e \), is a unique mathematical constant approximately equal to 2.71828. It is an irrational number, just like \( \pi \), meaning it can't be exactly written as a simple fraction.
Euler's number arises naturally in a variety of mathematical contexts, one of the most prominent being in the study of exponential growth and decay. It's a cornerstone in calculus, often appearing in formulas and equations involving rates of change.

• Euler's number is fundamental in continuous growth calculations, such as compound interest and population growth.
• It is the base of natural logarithms, commonly denoted as \( \ln \).
• In the expression \( e^3 \), Euler's number is raised to the power of 3. This means \( e \) is multiplied by itself twice.

Scientific Calculators

Scientific calculators are instrumental tools in higher mathematics and science, allowing users to perform complex computations easily. They come equipped with features specifically designed for tasks like evaluating exponential expressions. Most scientific calculators have a dedicated button for Euler's number, \( e \), which simplifies calculations involving this constant.

To evaluate expressions like \( e^3 \), follow these steps on a scientific calculator:

• Locate the \( e \) button, which might be directly labeled or accessed through a shift function.
• Enter the exponent using the '^' key or an equivalent symbol.
• Input the power, in this case, 3, and then press the '=' button to compute the expression.

Using a scientific calculator ensures high precision in results, vital for complex mathematical tasks.

Rounding Decimals

Rounding decimals is a crucial skill in mathematics, ensuring that you handle numbers appropriately when exact precision isn't necessary or practical. In problems requiring rounding to a specific decimal place, understanding the process helps maintain accuracy while simplifying the expression.

Here is how you can round to four decimal places:

• Identify the fourth decimal place in the number.
• Look at the fifth decimal place, which determines whether you round up or keep the number as is.
• If the fifth decimal is 5 or more, increase the fourth decimal place by one.
• If the fifth decimal is less than 5, leave the fourth decimal as it is.

Rounding ensures the result is manageable, especially when dealing with long decimal expansions like those found in solutions of exponential functions.