Problem 9 - Algebra 2

Question

Use a calculator to approximate each value to three decimal places. $$ \sqrt{77} $$

Step-by-Step Solution

Answer

The approximate value of \( \sqrt{77} \) is 8.775.

Step 1: Understand the problem

We need to find the square root of 77 and approximate it to three decimal places using a calculator.

Step 2: Input to Calculator

Enter the number 77 into the calculator and use the square root function to determine \( \sqrt{77} \).

Step 3: Record Raw Output

Once you press the square root button, the calculator will display a decimal value. For 77, this value should be approximately 8.774964387.

Step 4: Round to Three Decimal Places

To round to three decimal places, look at the fourth decimal place, which is 9 in this case. Since it's more than 5, round up the third decimal place from 4 to 5.

Step 5: Write the Final Answer

The rounded value of \( \sqrt{77} \) to three decimal places is 8.775.

Key Concepts

Calculator Usage

Calculators are incredibly useful tools for finding approximate values of square roots. They can save time and provide precise results quickly. When you need to find the square root of a number, such as 77, it's important to know how to properly use your calculator.
First, ensure that your calculator has a square root function. This is typically represented by a symbol that looks like a checkmark or by the operation \( \sqrt{} \).
To find \( \sqrt{77} \), begin by entering 77 into your calculator. Then, simply press the square root button.
The calculator will display a long decimal number, which represents the square root of 77. This process allows you to bypass complex arithmetic and focus on understanding the result.

Decimal Places

Decimal places refer to the number of digits shown to the right of the decimal point. When approximating, it’s common to use a specific number of decimal places, which helps us manage accuracy. For the square root of 77, we wanted an approximation to three decimal places.
After using the calculator, the raw output was approximately 8.774964387. The decimal portion after the whole number (8) is 7, 7, 4, and 9 in succession.
Focusing on three decimal places means we observe only the digits immediately following the decimal point up to the third position. In this case, they are 7, 7, and 4. This is crucial for ensuring the results meet the requirements for precision in calculations.

Rounding Numbers

Rounding is an essential skill when working with decimals, especially when calculators provide more digits than needed. To round a number, focus on the digit immediately following your cutoff point. If this digit is 5 or higher, round up the last digit you're keeping. If it's less than 5, simply keep the last digit as is.
For \( \sqrt{77} \), after we get 8.774964387, we want to round to three decimal places. Look at the fourth decimal place, which is 9. Since 9 is greater than 5, we increase the third decimal place by one, changing 4 to 5.
With these steps, the rounded value becomes 8.775. This practical approach ensures that your approximations are both simple and reliable, matching the desired accuracy.