How to Calculate the Square Root of 0.45 Using the Long Division Method

Calculating the square root of a decimal like 0.45 using the long division method involves several intricate steps. This guide will walk you through each step, ensuring you gain a clear understanding of the process.

Step-by-Step Guide

Step 1: Prepare the Number

Write 0.45 as 45/100. For simplicity, express it as 45 by moving the decimal point two places to the right. This adjustment will require adjusting the final result later.

Step 2: Group the Digits

Starting from the decimal point, group the digits in pairs. For 0.45, this results in:

Step 3: Set Up the Long Division

Find the largest number whose square is less than or equal to the first group (00). The largest number is 0 since
0^2 0. Write 0 above the line and subtract 0^2 0 from 00. This gives you 00.

Step 4: Bring Down the Next Group

Bring down the next group (45) next to the remainder (00) resulting in 0045.

Step 5: Double the Current Quotient

Double the quotient you have so far, which is 0:
2 times 0 0.
Write this down leaving a blank space next to it: 0_.

Step 6: Find the Next Digit

We need to find a digit x such that:
x leq 45
- Try x 6:
06 times 6 36 quadtext{which is less than 45}
- Try x 7:
07 times 7 49 quadtext{which is greater than 45}
- So the digit is 6.

Step 7: Update the Quotient and Remainder

Write 6 above the line, making the quotient 0.6.
Subtract 36 from 45:
45 - 36 9

Step 8: Bring Down the Next Group

Bring down the next pair of zeros (00) making the new number 900.

Step 9: Repeat the Process

Double the current quotient (0.6):
2 times 0.6 1.2.
Write this down: 1.2_.
Now find a digit y such that:
1.2y times y leq 900
- Try y 7:
1.27 times 7 8.89 quadtext{which is less than 900}
- Try y 8:
1.28 times 8 10.24 quadtext{which is greater than 900}
- So the digit is 7.

Step 10: Finalize the Calculation

Write 7 above the line, making the quotient 0.67.
Subtract 8.89 from 900:
900 - 889 11

Step 11: Continue for More Precision

You can continue this process to get more decimal places by bringing down more pairs of zeros and repeating the steps as needed.

Final Result

After completing these steps, you will find that the square root of 0.45 is approximately 0.67. For more precision, you can continue the long division process as outlined.