Problem 76 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to approximate the number. (Round to three decimal places.)\(24.7^{1.1}\)

Step-by-Step Solution

Answer

The result is approximately 32.396

Step 1: Compute the power

Use a calculator to compute \(24.7^{1.1}\).

Step 2: Round off the result

After calculating, round off the result to three decimal places.

Key Concepts

Calculator Usage

When dealing with complex mathematical operations like exponentiation, calculators are invaluable tools. These devices can quickly handle calculations that would otherwise take a significant amount of time if done manually. To calculate a number like \(24.7^{1.1}\), simply key in the base number, 24.7, then use the exponentiation function, often symbolized as '^' or labeled as 'exp' on the calculator, followed by the exponent, 1.1.
After inputting these values, the calculator will perform the complicated arithmetic behind exponentiation and present you with the result in an instant.
This handy tool saves time and minimizes errors in complex calculations, ensuring that your results are accurate. Remember to ensure your calculator is in the correct operational mode, typically Radian for trigonometric calculations unless specified otherwise.

Rounding Numbers

Rounding numbers is an important skill to master in mathematics, especially when precise values aren't necessary or when readings need to match a certain level of precision.
In our exercise, after computing \(24.7^{1.1}\) with a calculator, the result will likely have many decimal places.
To round to three decimal places:

• Identify the third decimal place.
• Look one digit beyond it.
• If this digit is 5 or above, increase the third decimal place by one.
• If it is less than 5, leave the third decimal place as it is.

This rounding convention helps make numbers easier to work with and understand, while still maintaining a reasonable degree of accuracy.

Approximate Calculations

Approximate calculations are useful when an exact number isn't required, and dealing with simplified figures is more feasible.
For many practical purposes, such as engineering or science, exactly knowing \(24.7^{1.1}\) might not be necessary to the nth decimal. In these cases, approximating values provides a fast and efficient way to handle numbers, retaining just enough accuracy for the task at hand.
By approximating the result of \(24.7^{1.1}\) to, for instance, three decimal places, you maintain a balance between simplicity and precision. This allows you to quickly gain insights from the calculations without getting bogged down by too much detail.
Always recognize the level of accuracy required in your work, and use approximation accordingly.