Problem 55 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest thousandth. $$\sqrt{87}$$

Step-by-Step Solution

Answer

The square root of 87, approximated to the nearest thousandth, is 9.327.

Step 1: Identify the Expression

We need to estimate the square root of 87, denoted as \( \sqrt{87} \). This expression requires the calculation of the non-perfect square number's root.

Step 2: Utilize the Calculator

To find \( \sqrt{87} \), enter 87 into your calculator and apply the square root function. Make sure your calculator is set to display several decimal places for greater accuracy.

Step 3: Read the Result

The calculator provides a result for \( \sqrt{87} \). Take note of the value displayed, including the decimal places.

Step 4: Approximate to the Nearest Thousandth

Round the result from the calculator to three decimal places, focusing on the thousandth digit. If the fourth decimal place (ten-thousandth digit) is 5 or greater, round up the thousandth place. Otherwise, keep it the same.

Key Concepts

Understanding Square Roots

The square root is a fundamental mathematical concept and an important one to grasp. When we talk about a square root, we're looking for a number which, when multiplied by itself, gives back the original number. For example, the square root of 9 is 3, since 3 multiplied by 3 equals 9. This is denoted as \( \sqrt{9} = 3 \).

When we deal with perfect squares, like 4, 9, 16, or 25, finding the square root is straightforward because these numbers have whole numbers as their roots. However, when the number is not a perfect square, like 87, things get a bit more complex, and that's where approximation comes into play.

Using Calculators Effectively

Calculators are powerful tools for tackling complex mathematical problems, especially those involving non-perfect squares. When approaching a problem like calculating \( \sqrt{87} \), the calculator becomes indispensable. Here's how you can make the most of it:

• Start by ensuring your calculator's display is set to show at least three decimal places. This is crucial for accuracy, especially when rounding to specific decimal places.
• Enter the number 87 and use the square root function, often depicted as \( \sqrt{x} \) or a similar symbol. This function automatically computes the square root of the input number.
• Carefully read the result. On most scientific calculators, it will display the root with several decimal digits, which allows for precise approximation.

Using your calculator effectively ensures that we get a reliable approximation for non-perfect squares.

Dealing with Non-Perfect Squares

Non-perfect squares are numbers that do not have an integer as their square root. Numbers like 87 fall into this category, making them a bit trickier to work with.

When calculating \( \sqrt{87} \), we do not end up with a neatly rounded number. Instead, we'll get a decimal, approximately 9.327. This is where understanding how to round becomes essential.

• Observe the digits after the decimal point. Consider up to three decimal places for your final answer.
• If the fourth decimal digit (the ten-thousandth place) is 5 or more, round the third digit up. For example, if your calculator shows 9.3276, you would round this to 9.328.
• If the fourth digit is less than 5, keep the third digit as it is, such as keeping 9.327 if the result was 9.3274.

Understanding these principles will help you accurately approximate roots of non-perfect squares!