Problem 39 - Prealgebra

Question

Use a calculator to find decimal approximations for each of the following numbers. $$\sqrt{72} \text { and } 6 \sqrt{2}$$

Step-by-Step Solution

Answer

\( \\sqrt{72} \approx 8.4853 \) and \( 6 \\sqrt{2} \approx 8.4852 \).

Step 1: Calculate the Square Root of 72

Use a calculator to find the square root of 72. Enter 72 and then the square root function, which should give you a decimal approximation of \( \sqrt{72} \approx 8.4853 \).

Step 2: Calculate 6 Times the Square Root of 2

Next, calculate \( 6 \sqrt{2} \). Using your calculator, first find the square root of 2, which is approximately 1.4142. Then multiply this value by 6 to get \( 6 \cdot 1.4142 \approx 8.4852 \).

Key Concepts

Decimal Approximation

Decimal approximation is a method used to express irrational numbers as decimals with a certain degree of precision. This is particularly useful when dealing with square roots or other complex mathematical operations where exact values are not easily represented. For example, the square root of 72 is an irrational number because it cannot be expressed as a simple fraction. Hence, using a decimal approximation allows us to use this number effectively in calculations.

• It involves rounding numbers to a specific number of decimal places. This enhances ease of use, especially in practical applications such as engineering and science.
• For instance, in Step 1 of the exercise, the square root of 72 is approximated as 8.4853. This number is precise enough for most calculations while not unnecessarily complex.

Decimal approximations require careful usage. The number of decimal places you choose might affect how accurate your calculations are.

Calculator Usage

Using a calculator is a practical way to simplify the process of finding decimal approximations for complex numbers like square roots. Calculators are equipped with functions that allow quick computation of operations that would otherwise be lengthy and error-prone by hand.
When finding the square root of 72 or calculating 6 times the square root of 2, follow these steps:

• Enter the number you want to find the square root for, such as 72.
• Press the square root button (√) to find its approximation, resulting in 8.4853 for 72 in Step 1.
• Similarly, for Step 2, first find the square root of 2 and then multiply the result by 6 using the multiplication button (*).

Using calculators ensures accuracy and saves time, providing instant results with minimized errors.

Mathematical Operations

Mathematical operations, such as addition, subtraction, multiplication, and division, play a crucial role in calculations requiring decimal approximations. Understanding these foundational concepts is essential, especially when dealing with complex problems involving square roots.

To approach a calculation like in Step 2, where we multiply a number by the square root of another number, follow these small yet crucial steps:

• First, identify the square root you'll need to operate on. For instance, the square root of 2 is approximately 1.4142.
• Then, use multiplication to scale this number as needed; for example, multiplying 1.4142 by 6 results in an approximation of 8.4852.

Such operations, although seemingly simple, form the basis of more advanced mathematical procedures, and understanding them ensures robust problem-solving skills.