Solve the Square Root of 0.24: Step-by-Step Guide with Detailed Explanation

Understanding the square root of 0.24 is a valuable skill that can be applied in various mathematical and scientific contexts. This guide will walk you through the process of calculating the square root of 0.24 both through traditional digit-by-digit approximation and by using a more straightforward fraction-based method. Follow along with detailed steps and explanations for a comprehensive understanding.

Traditional Digit-by-Digit Approximation Method

Calculating the square root of 0.24 manually can be done using the long division method or the digit-by-digit approximation technique. Here, we will outline the steps involved in this process:

Step 1: Set Up the Calculation

Start by placing a 0 followed by a decimal point on the answer panel. This is the initial guess for our square root.

Step 2: Break Down the Digits

Take the digits 24. The highest number whose square is less than or equal to 24 is 4, as 42 16. Place 4 after the decimal point on the answer panel.

Step 3: Calculate and Adjust

Place twice the number currently on the answer panel (2 x 4 8) on the left panel. Subtract 16 from 24 to get 8. Add a pair of 0’s to the right of 8, making it 800.

Step 4: Find the Next Digit

Identify the highest digit which, when placed to the right of 8 and multiplied by 80d, is less than or equal to 800. Here, 88 x 8 704 and 89 x 9 801. Therefore, 8 is suitable, and the answer panel now reads 0.48.

Repeat the process by calculating 2 x 48 96 and subtracting 704 from 800 to get 96. Add another 00 to 96, making it 9600.

Step 5: Continue the Process

Identify the highest digit d which when placed to the right of 96 and multiplied by 960d is less than or equal to 9600. Here, 969 x 9 8721 and 9609 x 9 86481. Therefore, 9 is suitable, and the answer becomes 0.489.

This process can be continued to achieve more precise results. The value of the square root of 0.24 as given by a calculator is approximately 0.489897949. To two decimal places, it is 0.49.

Fraction-Based Method for Simplification

For a more simplified and practical method, you can express 0.24 as a fraction and then take the square root of that fraction:

Step 1: Express as a Fraction

Write 0.24 as a fraction: 0.24 24/100.

Step 2: Simplify the Fraction

Divide both the numerator and the denominator by their greatest common factor, which is 4: 24/100 6/25.

Step 3: Take the Square Root of the Fraction

Take the square root of the simplified fraction: √(6/25) √6 / √25 √6 / 5.

Thus, the square root of 0.24 is approximately √6 / 5.

The traditional digit-by-digit method may seem more cumbersome, but it provides a deep understanding of how square roots are calculated. The fraction-based method, on the other hand, offers a more straightforward and simplified approach, making it easier to quickly determine the square root of a fraction.

Both methods are useful depending on the context. The fraction-based method is particularly useful for quick mental calculations, while the digit-by-digit method is valuable for a thorough understanding and manual calculations.