Problem 53 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest thousandth. $$\sqrt{12}+\sqrt{75}$$

Step-by-Step Solution

Answer

The expression approximates to 12.124.

Step 1: Calculate \(\sqrt{12}\)

Use your calculator to find the square root of 12. Enter 12 and press the square root button. You should get approximately \(\sqrt{12} \approx 3.464\).

Step 2: Calculate \(\sqrt{75}\)

Next, use your calculator to find the square root of 75. Enter 75 and press the square root button. You should get approximately \(\sqrt{75} \approx 8.660\).

Step 3: Add the Results

Now, add the two approximate values you obtained: \(3.464 + 8.660 = 12.124\).

Step 4: Round the Sum

Ensure the sum \(12.124\) is rounded to the nearest thousandth. The result is already rounded correctly to \(12.124\).

Key Concepts

Square Root Calculation

Calculating square roots is a fundamental concept in prealgebra that involves finding a number that, when multiplied by itself, results in the given value. For example, the square root of 12 is a value that when squared (multiplied by itself) gives 12. Using a calculator simplifies this process:

• Enter the number whose square root you want to find.
• Press the "square root" button, often depicted as a radical sign (√) on calculators.
• Read the result; this is the approximate square root of the number.

The principle of finding square roots is crucial as it frequently appears in equations and problems, especially when dealing with quadratic equations or measurements in geometry. Practicing this concept using a calculator helps students understand the magnitude of numbers and gain confidence in handling calculations efficiently.

Approximation

In mathematics, approximation is a technique that provides an estimated value close to the exact number. It's particularly useful when dealing with irrational numbers, which cannot be expressed as simple fractions. Square roots of non-perfect squares often result in irrational numbers, making approximation essential.

Here's how you can approximate using a calculator:

• Calculate the square root of a number using a calculator to get a decimal result.
• Use only the first few decimal places for practical purposes, preventing cumbersome computations.

For instance, to approximate \(\sqrt{12}\), you'd use 3.464 rather than expressing it with endless decimals. This reduces complexity while retaining a high level of accuracy. The goal with approximation is to provide a balance between precision and simplicity, especially in situations where exactness can be compromised for ease of computation.

Rounding to Thousandth

Rounding numbers to the nearest thousandth is a critical skill in ensuring results are both precise and easily interpretable. This concept requires understanding of decimal places, specifically knowing how to identify and manipulate the digits following the decimal point.

• The first digit after the decimal point is the tenth, the second is the hundredth, and the third is the thousandth.
• When rounding to the nearest thousandth, examine the fourth digit. If it is 5 or greater, increase the third digit by one.
• If the fourth digit is less than 5, the third digit remains unchanged.

In the solution provided in the exercise, the result is given as 12.124, which is already correctly rounded to the nearest thousandth. This precision ensures that all results maintain their integrity across mathematical computations, providing consistency and accuracy in problem-solving scenarios.