Problem 52 - Brief Applied Calculus

Question

Use a calculator to evaluate each expression. Round answers to two decimal places. $$ 5^{3.9} $$

Step-by-Step Solution

Answer

5^{3.9} ≈ 3043.29

Step 1: Input the Expression

Begin by turning on your calculator. Ensure your calculator is in normal calculation mode and ready to accept inputs. Enter the base of the expression, which is `5`, into the calculator.

Step 2: Apply the Exponent

Use the exponentiation function on your calculator. This is often represented by a `^` symbol or a button labeled `EXP`. After entering the base `5`, press the exponentiation button and then enter `3.9`, which is the power to which the base is raised.

Step 3: Calculate the Result

Once you have entered the expression as `5^3.9`, press the `=` or `Enter` button on your calculator to evaluate the expression. The calculator will compute the value, performing the power calculation.

Step 4: Round the Answer

The calculator will display a numerical result. Round this result to two decimal places as instructed. If the third decimal digit is 5 or greater, round the second decimal digit up.

Key Concepts

Using Exponentiation in Mathematics

Exponentiation is a mathematical operation involving numbers called the base and the exponent. The exponent refers to how many times the base is multiplied by itself. For instance, the expression \(5^{3.9}\) means that the number 5 is raised to the power of 3.9.
Understanding how to work with exponents is crucial because it's a basic building block in various areas of math, including algebra and calculus.

• The base (5) is the number being multiplied.
• The exponent (3.9) shows how many times the base is used as a factor.

This can get a bit tricky with non-integer exponents like 3.9, since it involves fractional power calculations. Luckily, calculators are designed to handle this complexity easily.Feel confident to regularly use calculators in such cases, as they help ensure precision.

Rounding Decimal Places

Rounding is the process of simplifying a number while retaining its value close to what it was. It's useful in keeping numbers more manageable, especially when they involve long decimal parts.
When you round a number to two decimal places, you look at the third decimal number to decide whether to round up or keep the second decimal as it is.
Some quick rules are:

• If the third decimal digit is between 0 and 4, leave the second decimal place as it is.
• If the third decimal number is 5 or more, increase the second decimal place by one.

By mastering rounding, you can ensure your answers are not only accurate but also easy to communicate and understand.

Power Calculations with a Calculator

Power calculations can be intimidating due to their level of complexity, but they become straightforward with a calculator. When faced with a problem like \(5^{3.9}\), using a calculator simplifies the process of calculating exponents.
Calculators have an exponent function, often marked as `^` or `EXP`. It allows you to input base values along with their exponents quickly. To enhance your calculator skills:

• Ensure the calculator is in standard mode before beginning your calculation.
• Double-check your inputs for accuracy before pressing `=`.
• Use rounding functions if the calculator doesn't display decimals immediately.

A calculator enhances the accuracy of power calculations, freeing you to focus on understanding and applying the results to broader problems.