Problem 39 - Algebra 2

Question

Use a calculator to approximate each value to three decimal places. $$ \sqrt{0.87} $$

Step-by-Step Solution

Answer

The square root of 0.87 is approximately 0.933.

Step 1: Understanding the Problem

We are asked to find a decimal approximation of the square root of 0.87 to three decimal places. This involves using either a calculator or an approximation method to determine \( \sqrt{0.87} \) with precision.

Step 2: Use a Calculator

To solve \( \sqrt{0.87} \), input it into a calculator, specifically ensuring that the calculator is set to display the result to at least three decimal places. Enter the square root function followed by 0.87.

Step 3: Record the Result

Once you calculate \( \sqrt{0.87} \), observe and record the result given by the calculator. Ensure the value obtained is expressed in three decimal places to meet the problem requirement.

Step 4: Verify the Precision

Check that the number displayed by your calculator, such as 0.9333, rounds correctly to three decimal places as needed. It's essential to note the fourth decimal to ensure proper rounding.

Key Concepts

Decimal Approximation

When we talk about decimal approximation, we're referring to the process of finding a number that is very close to a given mathematical expression. In most cases, we're dealing with numbers that are inherently irrational, like square roots, which don't have a simple or finite decimal representation. To manage this, we approximate the value to a certain number of decimal places. This process involves several steps:

• First, identify the number or expression you need to approximate. For instance, in the problem provided, that number is \( \sqrt{0.87} \).
• Next, decide the level of precision you need. The exercise specifies approximating to three decimal places, which gives us a balance between accuracy and simplicity.
• Lastly, use a reliable method or tool, like a calculator, to find this approximation.

This method is widely used in many fields, including engineering and physics, to simplify calculations and make numbers more manageable.

Calculator Usage

Using a calculator effectively is a crucial skill, particularly when working with complex expressions like square roots. For \( \sqrt{0.87} \), the first step is ensuring your calculator is set correctly:

• Confirm the calculator is in "normal" mode, not scientific or engineering, to prevent unexpected results.
• Set your calculator to show at least three decimal places, matching the problem's requirements.
• Enter the square root symbol or function (often a button labeled '√' or 'sqrt') followed by the number 0.87.

Once you've entered the expression properly, press "equals" or "enter" to get your result. Double-check that the display shows your answer to the required number of decimal places. This process helps ensure precision and correctness in your final answer.

Rounding Decimals

Rounding decimals is the final step in our approximation process, focusing on ensuring the result meets the required precision. When rounding to three decimal places, as specified in the exercise, consider the following:

• Identify the third decimal number. In our example, this might initially be 0.9333 when examining \( \sqrt{0.87} \).
• Look to the fourth decimal place to determine if the third place rounds up or stays the same. If the fourth decimal is 5 or higher, round the third place up. Otherwise, leave it unchanged.
• Record the rounded number accurately. For a result like 0.9333, rounding to three decimals results in 0.933 because the fourth decimal was 3.

This method of rounding ensures your answer aligns with standard precision norms, giving you a result that is not only mathematically accurate but also consistent with the exercise's requirements.