Problem 1 - Algebra 2

Question

Use a calculator to evaluate each expression to four decimal places. \(e^{6}\)

Step-by-Step Solution

Answer

\(e^{6} \approx 403.4288\).

Step 1: Understanding the Expression

The expression given is \(e^{6}\), which means we need to calculate the value of the mathematical constant \(e\) raised to the power of 6. The constant \(e\) is approximately 2.71828, which is a well-known irrational number.

Step 2: Using a Calculator

To find \(e^{6}\), you will need a scientific calculator capable of exponential calculations. Most calculators have an \(e^x\) or \(exp(x)\) function that you can use. Input \(6\) as the exponent.

Step 3: Calculating the Result

Evaluate \(e^{6}\) using the calculator's exponential function. Ensure that your calculator is set to display results to four decimal places.

Step 4: Recording the Answer

Once you have the result from the calculator, note it down as: \(e^{6} \approx 403.4288\). This value is rounded to four decimal places as required.

Key Concepts

Irrational Numbers

An irrational number is a number that cannot be expressed as a simple fraction. This means it cannot be written as a ratio of two integers. Instead, it has a decimal expansion that goes on forever without repeating. Examples of irrational numbers include the square root of 2, π (pi), and the base of natural logarithms, e.

In mathematics, irrational numbers are important because they help us describe numbers that are not easily expressed in simpler terms. The number referred to in your exercise, e, is a famous irrational number. It is approximately 2.71828, but its decimal expansion continues infinitely without repetition. Knowing about irrational numbers helps us understand why certain values cannot be exactly calculated, only approximated.

Scientific Calculators

A scientific calculator is a type of calculator designed to handle advanced mathematical functions that regular basic calculators cannot. It is an essential tool for tackling complex equations in mathematics and science.

Here are some features you might find in a scientific calculator:

• Functions for exponential calculations, such as the button labeled \(e^x\) or \(\text{exp}(x)\).
• Ability to handle trigonometric functions like sine and cosine.
• Logarithmic functions, typically written as \(\log\) and \(\ln\).
• Options for switching between different number systems and representations.

To use it effectively, you should become familiar with its layout, especially if you need to calculate expressions like \(e^6\), which involves using the exponential function feature.

Rounding to Decimal Places

Rounding to decimal places involves adjusting a number so that it has a specific number of digits after the decimal point. This is particularly important when presenting results, as it ensures consistency and accuracy without displaying unnecessary information.

Here's how you can round a number to four decimal places:

• Identify the fourth decimal place.
• Check the digit immediately after it (the fifth decimal place).
• If this digit is 5 or greater, increase the fourth decimal place by 1.
• If it's less than 5, leave the fourth decimal place as it is.

For example, in the exercise solution, the calculation of \(e^6\) resulted in 403.428752, which was rounded to 403.4288. This approach provides an accessible level of precision needed for many scientific applications.

Mathematical Constants

In mathematics, constants are values that do not change. Mathematical constants are important, as they are universal across equations and applications. One of the most famous constants is e, used extensively in calculus, complex numbers, and statistics.

Some well-known mathematical constants include:

• e \(\approx\) 2.71828, which is known as Euler's number.
• π \(\approx\) 3.14159, representing the ratio of a circle's circumference to its diameter.
• \(\phi\) \(\approx\) 1.61803, known as the golden ratio.

These constants appear in various forms and are fundamental to many mathematical proofs and principles. The constant e is particularly vital in growth models, lending itself to applications in finance, biology, and physical sciences.

Power of a Number

The power of a number is an expression like \(x^n\), where x is the base and n is the exponent. This concept involves repeating multiplication, similar to how we described the exercise dealing with \(e^6\).

The basics of powers include:

• When n is positive, you multiply the base by itself n times.
• If n is zero, any non-zero base raised to this power equals 1.
• When n is negative, it represents the reciprocal of the base raised to the absolute value of n.

Understanding powers is crucial in many areas of mathematics, like algebra and calculus. They allow us to compactly express operations that would otherwise be lengthy to write out or compute.