Problem 56 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest thousandth. $$\sqrt{68}$$

Step-by-Step Solution

Answer

The square root of 68, rounded to the nearest thousandth, is approximately 8.246.

Step 1: Understand the Problem

We need to find the square root of 68 and round it to the nearest thousandth.

Step 2: Use the Calculator

Enter 68 into the calculator and press the square root (√) function to determine the square root. Confirm that the result appears.

Step 3: Record the Calculator Result

The calculator shows the square root of 68 as approximately 8.246211251.

Step 4: Round to the Nearest Thousandth

To round 8.246211251 to the nearest thousandth, focus on the first three decimals: 8.246. Check the fourth decimal place, if it is 5 or more, increase the third decimal by 1.

Step 5: Apply Rounding Rules

Since the fourth decimal digit is 2, which is less than 5, we don't need to round up. Thus, the rounded value remains 8.246.

Key Concepts

Square Root Calculation

Calculating the square root of a number can be daunting at first, but it's easier when you break it down into simple steps. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 68 is a number that, when squared, equals 68. This number isn't a neat whole number, but that's where calculators come in handy. - To calculate the square root of 68, begin by locating the square root function on your calculator, often symbolized as "√". - Input the number 68 and press the square root function to perform the calculation. - The calculator processes the input and calculates an approximate value because 68 doesn't have a perfect square root like numbers 64 and 81 do. The square root of 68 results in a decimal which can be further refined by rounding it to make it more practical for certain measurements or reports.

Decimal Places

Decimal places are integral when dealing with precise calculations, especially in scientific and mathematical contexts. A decimal place in a number refers to the digits that appear after the decimal point. - When a decimal number like 8.246211251 is present, each digit after the decimal contributes to the number's precision. - The first decimal digit is the tenths, the second is the hundredths, and the third is the thousandths. In rounding, focusing on specific decimal places allows for accuracy control. For the example of 8.246211251 in the context of rounding to the nearest thousandth, the process involves: - Looking at the first three decimal places (8.246). - Checking the digit right after the thousandth (fourth decimal place), which decides if the thousandth place goes up or stays the same. Understanding decimal places is crucial in rounding, ensuring numbers are accurate yet manageable.

Calculator Usage

Using calculators in mathematical operations saves time and reduces arithmetic errors. Familiarity with a calculator's functions is essential for precise calculations like square roots. Here’s how to effectively utilize a calculator: - **Identify the Function Buttons:** Recognize buttons like "sqrt" or "√" for finding square roots. - **Input Numbers Accurately:** Ensure you enter numbers correctly; one wrong press can lead to incorrect results. - **Review the Display:** Always check the calculator's screen after performing an operation to confirm the expected result is shown. Calculators may vary across brands and designs, but the core functions remain similar. Practice using your calculator regularly to boost your confidence in finding results like those for square roots or working with decimals. Proper usage ensures your calculations are not only fast but also reliable.