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Chapter 5: Taking Square Root
F.2 Mathematics Enrichment
Taking Square Root
Taking Square Root of a number
can be done easily by using a calculator in normal
lessons. How about when there is no calculator?
This is common in competitions. Do we have any
way to find the square root without using a
calculator?
1.
Simple Square Numbers
12 =
62 =
112 =
162 =
2.
22 =
72 =
122 =
172 =
32 =
82 =
132 =
182 =
42 =
92 =
142 =
192 =
52 =
102 =
152 =
202 =
Square Root of a Large Square Number
Find
676 (Given that
676 is an integer).
Step 1: Find the range of the square root. As 252 = 625 and 302 = 900,
25  676  30
Step 2: Find the unit digit of the square root.
Unit digit of the square number
Unit digit of the square root
0 1 4 9 6 5 6 9 4 1
0 1 2 3 4 5 6 7 8 9
From the table, 676  26
e.g.1: Given that 1764 and 3249 are both square numbers, find their square roots.
1600 = __________
2500 = __________
_52 = ___25
2500 = __________
3600 = __________
_52 = ___25
_____ < 1764 < ______
_____ < 3249 < ______
By considering the unit digits,
1764  __________
3249  __________
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Chapter 5: Taking Square Root
F.2 Mathematics Enrichment
3.
Square Root of Any
Degree of Accuracy
Find
Number
with
a
given
650.5 . (correct to the nearest integer)
Step 1: Find the range of the square root. As 252 = 625 and 262 = 676,
25  650.5  26
Hence, 650.5 is either 25 or 26.
Step 2: Find the square of the average of two numbers got in Step 1.
25.52 = 650.25,
as 650.25 < 650.5,
25.5  650.5  26
As a result, 650.5  26 (correct to the nearest integer)
e.g.2: Find the square root of 1234. (correct to the nearest integer)
900 = __________
1600 = __________
1225= __________
362 = __________
Therefore, __________  1234  __________
__________2 = __________
__________  1234  __________
As a result, 1234  __________
e.g.3: Find the square root of 56.78. (correct to 1 decimal place)
49 = __________
64 = __________
56.25 = __________
7.62 = __________
Therefore, __________  56.78  __________
__________2 = __________
__________  56.78  __________
As a result, 56.78  __________
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Chapter 5: Taking Square Root
F.2 Mathematics Enrichment
4.
“Long Division” to Find Square Root
Find
650.5 . (correct to the nearest integer)
2
6
- 4
2
2
50.
50
Step 1: Separate the number into groups of 2 digits
(starting from the decimal point): 6, 50, 50
(add a zero following the last ‘5’).
50
Step 2: Look for an integer such that its square is
just smaller than the first group (6), and the
integer is 2. Now we have a remainder 250.
2
4__
2
45
2
6
- 4
2
__.
50.
2
6
- 4
2
- 2
5.
50.
50
50
25
25
50
50
Step 3: Then double what we have in the result
(2 x 2 = 4).
50
Step 4: Fill in the blanks in step 3 by the same
digit such that the product of (4x) and x is
just smaller than the remainder 250. The
digit should be 5.
Step 3 and 4 can then be repeated to obtain the required degree of accuracy. (00 may be
added if the next group of two digits is needed)
2
45
50__
2
6
- 4
2
- 2
5.
50.
50
25
25
__
50
2

50
45
2
6
- 4
2
- 2
505
-
5.
50.
50
25
25
25
5
50
50
25
25
650.5  25.5...
 650.5  26 (correct to the nearest integer)
Use the “Long Division” method to find the following square roots.
e.g.4: Find the square root of 1234. (correct to the nearest integer)
e.g.5: Find the square root of 56.78. (correct to 1 decimal place)
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Chapter 5: Taking Square Root
F.2 Mathematics Enrichment
5.
How Does The “Long Division” Work?
Remember the identity (a + b) 2 = a2 + 2ab + b2?
To explain the method, we change the identity a little bit:
(10a + b)2 = 100a2 + 20ab + b2
Take
676 as an example.
a
6
a
2
46
2
6
- 4
2
- 2
We want to find the largest integers a, b such that (10a + b)2
76 is just less than or equal to 676.
We test for a first, obviously, the largest possible value is 2.
6
76
Then what’s left = 676 – 400 = 276
Now, we have to try the greatest b such that
20(2)b + b2 = (40 +b) b is just less than or equal to 276
76
76 As 46(6) = 276, 676  26
Use methods in section 3 or 4 to find the following square roots.
(correct to 2 sig. fig.)
2
38
50
35.9
Page 4
Chapter 5: Taking Square Root
F.2 Mathematics Enrichment
Homework Solve the following questions.
1.
Given that 7921 and 15129 are square numbers, find their square roots.
2.
Use methods in section 3 or 4 to find the following square roots. (correct to 2 sig. fig.)
(a) 12345
(b) 187.69
(c) 35.835
3.
The figure shows an acute-angled triangle with area 120.
Find the integer closest to x.
[6th Pui Ching Invitational Mathematics Competition
Heat Event (Secondary 2), no.13]
Hint: Cut the triangle into two right-angled triangles by adding a suitable height
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