Find the square root of the following number by division method: 186624

To find the square root of 186624 using the division method, follow these steps:

Step-by-Step Process

1. Group the Digits

Start by grouping the digits of the number from right to left in pairs. For 186624, the pairs are:

• 18 | 66 | 24

2. Find the Largest Square

Identify the largest square number less than or equal to the first group (18). The largest square is 3 (since 3 x 3 = 9).

Write 3 above the line and subtract 9 from 18:

• 18 - 9 = 9

3. Bring Down the Next Pair

Bring down the next pair (66) next to the remainder:

• 9 | 66 becomes 966

4. Double the Quotient

Double the quotient (3) to get 6. Now, find a digit (x) such that (60 + x) x x is less than or equal to 966.

Testing x = 1: (60 + 1) x 1 = 61, which is too low.

Testing x = 2: (60 + 2) x 2 = 124, still low.

Testing x = 3: (60 + 3) x 3 = 189, still low.

Testing x = 4: (60 + 4) x 4 = 256, still low.

Testing x = 5: (60 + 5) x 5 = 325, still low.

Testing x = 6: (60 + 6) x 6 = 396, still low.

Testing x = 7: (60 + 7) x 7 = 441, still low.

Testing x = 8: (60 + 8) x 8 = 496, still low.

Testing x = 9: (60 + 9) x 9 = 585, still low.

Testing x = 10: (60 + 10) x 10 = 700, still low.

Testing x = 11: (60 + 11) x 11 = 781, still low.

Testing x = 12: (60 + 12) x 12 = 864, still low.

Testing x = 13: (60 + 13) x 13 = 961, which works!

Write 13 above the line next to 3, making it 43. Subtract 961 from 966:

• 966 - 961 = 5

5. Bring Down the Next Pair

Bring down the next pair (24) to get 524.

6. Repeat the Process

Double the quotient (43) to get 86. Now find a digit (y) such that (860 + y) y is less than or equal to 524.

Testing y = 0: (860 + 0) x 0 = 0, too low.

Testing y = 1: (860 + 1) x 1 = 861, too high.

So, y = 0 works. Write 0 above the line, making it 430.

Subtract 0 from 524:

• 524 - 0 = 524

Final Result

The square root of 186624 is approximately 432.