Problem 42 - Prealgebra

Question

Use a calculator to work. Approximate each of the following square roots to the nearest ten thousandth. $$\sqrt{12.5}$$

Step-by-Step Solution

Answer

The square root of 12.5, rounded to the nearest ten-thousandth, is approximately 3.5355.

Step 1: Enter the Square Root Function

To approximate the square root of 12.5, first, turn on your calculator. The function to find a square root is usually indicated by the symbol  under the function's name, such as 'sqrt'. Enter '12.5' and then press the square root button.

Step 2: Calculate the Approximate Value

After entering and executing the square root function for 12.5, the calculator will display a decimal number. Write down this number as it appears. This number is the unrounded result of the square root of 12.5.

Step 3: Rounding the Result

Round the number obtained from the calculator to the nearest ten-thousandth (four decimal places). First, identify the fifth decimal place. If this digit is 5 or greater, increase the fourth decimal place value by one. If it's less than 5, keep the fourth decimal figure as it is. Write down the rounded result.

Key Concepts

Calculator Usage for Square Roots

Using a calculator can greatly simplify the process of finding square roots, especially when dealing with non-perfect squares like 12.5. These devices are equipped with a specific function for calculating square roots, often marked by the symbol \(\sqrt{}\), to quickly provide you with an approximate decimal value.

When you need to approximate square roots, here’s a streamlined procedure to follow with your calculator:

• Firstly, turn on your calculator and ensure it is in the standard mode.
• Identify and press the square root function, which might be accessed directly or by using a shift key or similar.
• Enter the number you need the square root of, in this case, 12.5.
• Press the equals or enter key to obtain the approximate result.

It’s straightforward and ensures accuracy, letting the calculator do the complex computations for you, so you can focus on understanding the approximate value you get.

Decimal Rounding

Once you obtain a decimal result from your calculator, the next step is rounding it to the desired place. Rounding to the nearest ten-thousandth is a common requirement in many math problems. This ensures precision while maintaining simplicity. Let’s break down how to round decimals step by step:

• First, identify the digit at the fourth decimal place, since you want to round to the nearest ten-thousandth.
• Next, look at the fifth decimal place, as this will influence the rounding of the fourth.
• If the fifth digit is 5 or more, increase the fourth digit by one.
• If the fifth digit is less than 5, keep the fourth digit unchanged.

By following these simple steps, you can adjust the initial decimal result from your calculator to better fit the required level of detail. This helps in situations where precision is paramount, especially in scientific contexts or detailed financial calculations.

Prealgebra Concepts

Understanding prealgebra concepts is crucial as they lay the groundwork for more advanced mathematical topics. Among these foundational ideas are operations with square roots, which initially may seem complex but are quite manageable with the right tools and approach.

• Square roots are essentially the inverse function of squaring a number.
• They are especially useful in solving equations where unknown variables might have been squared in the problem setup.
• Mastering this entails recognizing that only non-negative numbers will typically result in real square roots.

Prealgebra paves the way to algebra and calculus, by introducing students to logical thinking and problem-solving techniques. So, understanding how to approximate square roots and manipulate decimal values through rounding processes forms an integral part of a student’s journey in mathematics, enabling a smoother transition to more advanced topics.