Understanding Unit Digits
The last digit of a perfect square can tell you the last digit of its square root.
| If a number ends in... | ...its square root must end in: |
|---|---|
1 | 1 or 9 |
4 | 2 or 8 |
9 | 3 or 7 |
6 | 4 or 6 |
5 | 5 |
0 | 0 |
Perfect squares never end in 2, 3, 7, or 8.
Method: Finding Square Roots of Perfect Squares
A fast estimation method for 4 and 5-digit numbers.
Example: Find √1521
1
Look at the unit digit
The number
1521
ends in 1
. So, the square root must end in 1
or 9
.2
Ignore the last two digits
Ignore
21
. We are left with 15
.3
Find the nearest smaller perfect square
The largest perfect square less than or equal to
15
is 9
, which is 3²
. This means the tens digit of our answer is 3.4
Form the two possibilities
From steps 1 and 3, our possible answers are
31
or 39
.5
Find the correct one
Multiply the tens digit (
Since our remaining number from Step 2 (
3
) by the next consecutive number (4
): 3 × 4 = 12
.Since our remaining number from Step 2 (
15
) is greater than 12, we choose the larger of our two possibilities.Result
The answer is 39.
Example: Find √2704
1
Unit Digit
Ends in
4
, so root ends in 2
or 8
.2
Remaining Number
Ignore
04
, leaving 27
.3
Tens Digit
Largest square ≤ 27 is
25 (5²)
. So, tens digit is 5.4
Possibilities
The answer is either
52
or 58
.5
Check
Multiply tens digit by next number:
5 × 6 = 30
. Since 27
is less than 30, we choose the smaller possibility.Result
The answer is 52.
Example: Find √3969
1
Unit Digit
Ends in
9
, so root ends in 3
or 7
.2
Remaining Number
Ignore
69
, leaving 39
.3
Tens Digit
Largest square ≤ 39 is
36 (6²)
. So, tens digit is 6.4
Possibilities
The answer is either
63
or 67
.5
Check
Multiply tens digit by next number:
6 × 7 = 42
. Since 39
is less than 42, we choose the smaller possibility.Result
The answer is 63.
Example: Find √7744
1
Unit Digit
Ends in
4
, so root ends in 2
or 8
.2
Remaining Number
Ignore
44
, leaving 77
.3
Tens Digit
Largest square ≤ 77 is
64 (8²)
. So, tens digit is 8.4
Possibilities
The answer is either
82
or 88
.5
Check
Multiply tens digit by next number:
8 × 9 = 72
. Since 77
is greater than 72, we choose the larger possibility.Result
The answer is 88.
Example: Find √12544
1
Unit Digit
Ends in
4
, so root ends in 2
or 8
.2
Remaining Number (for 5-digit numbers)
Ignore
44
, leaving 125
.3
First Digits
Largest square ≤ 125 is
121 (11²)
. So, first part is 11.4
Possibilities
The answer is either
112
or 118
.5
Check
Multiply first part by next number:
11 × 12 = 132
. Since 125
is less than 132, we choose the smaller possibility.Result
The answer is 112.