Problem 49 - Applied Calculus

Question

\(49-52 .\) Use a calculator to evaluate each expression. Round answers to two decimal places. $$ 7^{0.39} $$

Step-by-Step Solution

Answer

\(7^{0.39} \approx 3.05\)

Step 1: Understand the Expression

We need to evaluate the expression \(7^{0.39}\) using a calculator. The base is 7, and the exponent is 0.39. This indicates that we need to find the 0.39th power of 7.

Step 2: Input the Expression into the Calculator

Turn on the calculator and enter the base value, which is 7. Then use the exponentiation function (often marked as '^' or 'EXP') and enter 0.39 as the exponent.

Step 3: Calculate the Result

After entering the expression into the calculator correctly as \(7^{0.39}\), press the '=' or 'Enter' key to compute the result. The calculator will output the value of raising 7 to the power of 0.39.

Step 4: Round the Answer

Look at the calculator's output, which is the unrounded result of \(7^{0.39}\). Round this result to two decimal places to get the final answer.

Key Concepts

Calculator Usage

Using a calculator to perform exponentiation is a handy skill, especially when dealing with non-integer exponents like in the expression \(7^{0.39}\). Calculators offer various functions, and knowing how to navigate these features is crucial. To evaluate an exponentiation like \(7^{0.39}\), you first need to:

• Ensure your calculator is on. This may sound basic, but it helps eliminate any initial confusion.
• Enter the base number, which in our case is 7.
• Locate the exponentiation function, often represented as "^" or "EXP". This function allows you to raise the base to any power.
• Input the exponent, here it is 0.39.
• Press the equals '=' or 'Enter' to compute the calculated result.

Using these steps ensures that you can efficiently perform calculations with accuracy on your calculator, saving time and reducing errors.

Rounding Numbers

Rounding numbers is crucial for simplifying results and for representation purposes, especially when working with decimals. In scientific calculations, rounded numbers help in presenting an easy-to-understand answer. When a number needs to be rounded to two decimal places:

• Identify the digit in the third decimal place.
• If this digit is 5 or higher, round up the second decimal place by adding one to it.
• If it’s less than 5, leave the second decimal place as it is.
• Remove all digits following the second decimal place.

Using this method, you ensure that your mathematical results are precise and presented clearly, even when working with more complex numbers like those resulting from exponentiation.

Decimal Places

Decimal places refer to the numbers after the decimal point in a decimal number. Understanding them is key to precision in math. For example, in the number 3.456, the digits 4, 5, and 6 are in the first, second, and third decimal places, respectively.Choosing how many decimal places to use depends on the required precision:

• Two decimal places are commonly used for financial calculations, offering a balance between accuracy and simplicity.
• More decimal places are typically used in scientific calculations to reduce rounding errors.

Ensuring the correct number of decimal places helps maintain accuracy and clarity in mathematical results, particularly in exercises such as evaluating \(7^{0.39}\) and rounding outcomes effectively.