Problem 38 - Algebra 2

Question

Use a calculator to evaluate each expression to four decimal places. $$ e^{3} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

The expression we need to evaluate is \( e^{3} \). The constant \( e \) is known as Euler's number, approximately equal to 2.71828. Our task is to raise this constant to the power of 3 and evaluate the result to four decimal places.

Step 2: Calculating with a Calculator

Use a scientific calculator to find the value of \( e^{3} \). Ensure the calculator is set to the correct mode for exponential calculations. Input \( e \), usually represented as a button on scientific calculators, and then press the power function (often denoted as \( ^{ } \) or with a caret symbol \(^\wedge\)) and enter 3.

Step 3: Rounding the Result

Once the calculator provides the answer, check the display for the value of \( e^{3} \). The calculator should show something like 20.08553692. Round this number to four decimal places to obtain the final result, which is 20.0855.

Key Concepts

Exponential Functions and Euler's Number

Exponential functions are a key concept in mathematics where a constant base is raised to a variable exponent. One of the most important constants used in these functions is Euler's number, denoted as \( e \). Euler's number is an irrational number, meaning it cannot be accurately expressed as a simple fraction. Its approximate value is \( 2.71828 \), but it continues indefinitely without repeating.
The exponential function \( e^x \) is used in various fields such as biology, economics, and physics due to its unique properties, like continuous compound interest calculation and growth processes modeling. In our exercise, we focused on evaluating \( e^3 \). Understanding these properties helps in practical applications and theoretical problems alike.
Whenever you evaluate an expression such as \( e^x \), ensure clarity over what the exponent represents in the context of your problem. This understanding is crucial because the behavior of the exponential function changes significantly with different exponents.

Using a Scientific Calculator for Exponential Functions

A scientific calculator is a handy tool for evaluating exponential functions, especially when dealing with complex constants like Euler's number. Most scientific calculators have a dedicated button for \( e \), making it straightforward to perform calculations like \( e^3 \).
To compute \( e^3 \) using a scientific calculator, you will typically follow these steps:

• First, locate the \( e \) button, which is sometimes shared with the logarithmic function ln.
• Press the exponentiation button, which might look like \( ^{ } \) or use a caret symbol \(^\wedge\).
• Enter the exponent, in this case, 3, and then hit the equal sign to execute the operation.

Make sure your calculator is functioning correctly in its scientific mode to handle these types of calculations effectively. This functionality is crucial for precision, which is often required in academic and professional settings.
Familiarity with all available functions on your scientific calculator allows you to leverage their full potential and save time on complex calculations.

Rounding Numbers in Calculations

Rounding is an essential skill in mathematics and most scientific calculations to maintain clarity and usability of results. When you round a number, you are essentially finding an approximation that is "close enough" for your intended purpose.
In our exercise with \( e^3 \), we needed to round the result to four decimal places. Begin by identifying the digit in the fourth decimal place. Also, look at the digit just after it (the fifth decimal place) to determine whether to round up or leave unchanged.
For example, if your calculated result was 20.08553692, you would keep 20.0855 since the fifth decimal place "3" does not require rounding up the fourth decimal digit. Rounding provides a clean and concise answer suitable for most requirements except in very precise scientific investigations.
Use these guidelines:

• If the digit after your desired decimal place is 5 or greater, round up.
• If less than 5, leave the last desired digit as it is.

Rounding makes numbers more manageable and is a fundamental process in reporting scientific data, completing financial analyses, and much more.