Problem 14 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression to four decimal places using a calculator. $$e^{6}$$

Step-by-Step Solution

Answer

The expression \(e^{6}\) evaluates to approximately 403.4288 when rounded to four decimal places.

Step 1: Identify the key parameters

The key parameters in this exercise are the number \(e\) (Euler's number) and the power to which it is raised, which is 6.

Step 2: Input the expression into the calculator

Enter the expression \(e^{6}\) into a scientific calculator. Most calculators have a dedicated button for the constant \(e\). You will likely find it as a secondary function of a button (usually requires pressing the '2nd' or 'shift' button first). After inputting \(e\), press the exponentiation button (usually shown as '^') and then input '6'.

Step 3: Evaluate the expression

Press the equals button on the calculator to get the result.

Step 4: Round to four decimal places

Scientific calculators typically display results with many decimal places. Round your result to the 4th decimal place by looking at the 5th digit. If it is 5 or greater, you round up the 4th decimal place by one. If it's less than 5, you leave the 4th decimal place as it is.

Key Concepts

Euler's Number

Euler's number, often symbolized as e, is a mathematical constant approximately equal to 2.71828. This irrational number is uniquely important in mathematics, particularly in calculus, as the base of natural logarithms. It occurs naturally in various growth and decay problems, such as compound interest and population models.

When evaluating expressions like e^{6}, you're looking at the exponential function with base e, which reflects continuous growth. The expression e^{x} means e raised to the power of x. In our original exercise, x equals 6. So, we want to find the value of e when it is multiplied by itself 6 times. This gives us an insight into exponential growth at a specific rate, and how it scales over time.

Scientific Calculator Use

A scientific calculator is an essential tool in mathematics, chemistry, physics, and engineering. Its ability to handle a wide array of functions, including exponentials, logarithms, trigonometry, and more, makes it invaluable.

For our exercise, the scientific calculator lets us compute e^{6} easily. These calculators have a dedicated button for e, often labeled as EXP or accessible by hitting a 2nd or shift function key. After pressing this, input your exponent value and the calculator does the complex computation instantaneously. This tool effectively cuts down on calculation time and helps avoid errors that can occur when dealing with multiple-digit numbers or complex functions.

Rounding Decimal Places

Rounding decimals is a common mathematical practice to simplify numbers while maintaining an appropriate level of accuracy. To round a number to four decimal places, you look at the fifth decimal place. If this digit is 5 or higher, you increase the fourth decimal place by one. If it’s 4 or lower, the fourth decimal place remains unchanged.

In our example, after evaluating e^{6} on the calculator, we round the result to four decimal places. This step is crucial as it ensures the precision needed for the problem, without retaining unnecessary and potentially confusing additional digits. It's a way of balancing detailed accuracy with practical simplicity.