Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the expression. Round your result to three decimal places.\(e^{2 / 3}\)

Step-by-Step Solution

Answer

The value of \(e^{2 / 3}\), rounded to three decimal places, is 1.947.

Step 1: Input the expression into the calculator

Firstly, access a calculator that has an exponent and the mathematical constant \(e\). Enter the expression \(e^{2 / 3}\) into the calculator.

Step 2: Calculate

Press the 'equal' button to perform the calculation. The calculator will provide a result.

Step 3: Rounding the result

The expression \(e^{2 / 3}\) will likely provide a number with many decimal places. Round this number to three decimal places. Remember that if the digit at the fourth decimal place is greater than or equal to 5, increase the third digit by one.

Key Concepts

Rounding Numbers

When working with numbers, especially in mathematical calculations or scientific measurements, rounding is a common practice. It helps to simplify numbers and make them easier to work with. In this example, we are asked to round to three decimal places. But how exactly do we do this?

• First, look at the number in question. Suppose we have a number like 0.94372.
• Identify the third decimal place, which is 3 in this case. Then, check the digit right after it (fourth decimal place), which is 7 here.
• If this next digit is 5 or greater, you increase the third digit by 1. Thus, 0.94372 becomes 0.944.
• If the next digit is less than 5, simply leave the third decimal digit as it is.

Remember, rounding helps in truncating unnecessary precision while ensuring that calculations remain manageable and understandable.

Using a Calculator

Calculators are incredibly handy tools that help make complex mathematical calculations easier and quicker. When dealing with expressions or operations like finding the value of a power of a number, calculators are indispensable.

• Ensure your calculator has the capability to handle exponents and constants like e. Most scientific calculators will have an "e^x" button.
• To evaluate an expression like \(e^{2/3}\), you will typically first type "e," followed by the exponent, which in this case is 2/3.
• Once the expression is correctly inputted, press the equals button. The calculator will instantly compute the result.
• For precision, ensure that you are familiar with your calculator's functions to avoid errors in input.

Using calculators efficiently requires understanding its function keys and practicing regularly with different types of expressions.

Mathematical Constant e

The mathematical constant \(e\) is an important number in mathematics, often used in calculus and complex calculations. It is known as the base of natural logarithms and has a value approximately equal to 2.71828.

• \(e\) is an irrational number, which means it cannot be expressed as a simple fraction.
• Just like \(\pi\), \(e\) has an infinite number of non-repeating decimals.
• This constant is widely used in various contexts such as exponential growth, compound interest calculations, and in calculating derivatives in calculus.

Understanding \(e\) can provide insights into natural processes, as it is often used to model real-world phenomena.