Download Square and Square root

Square and Square root
Square = a number multiply by itself
Example: Square of 4 = 4x4 = 16
In general, the square of a number, say 4, is written as 42.
The square of a number also gives the area of a Square, which has 4 equal sides.
Exercises:
1. Find the square for each of the following numbers:
6, 12, 15, 21, 35, 68.
2. Find the square for each of the following numbers:
10, 20, 30, 100, 200, 900, 1000
Observed that the squares of any multiples of tens, hundreds, thousands, etc. can be easily obtained by
a) get the square of the leading non-zero digit(s), e.g. the number 7 in 700 and the number 13 in
1300, and
b) add 2 times the number of zeros after the answer obtained in a).
Example: To get the square of 800
a) the leading non-zero digit is 8, so 8x8 = 64
b) in 800, there are 2 zeros in the number, so we have to add 2x2=4 zeros after 64, and hence
8002 = 640000.
To get the square of 240
a) the leading non-zero digits are 24, so 24x24 = 576
b) in 240, there is 1 zero in the number, so we have to add 2x1=2 zeros after 576, and hence
2402=57600
Exercise
1. Find the square for each of the following numbers:
70, 120, 300, 5600
1
Square roots
Let b=a2. Now, if you are given b, and you would like to find a, this is called finding the square root of b,
or √𝑏.
Exercise
1. Find the square root of each of the following:
a) √4 × 4
b) √7 × 7 × 13 × 13
c) √2 × 2 × 3 × 3 × 5 × 5
Observed from the above exercises that
√7 × 7 × 13 × 13 =
7 × 13 × 7 × 13 =
(7 × 13)2 = 7 × 13
Hence, to find the square root of a number, one way is to find all the factors of the number. Put those
numbers with the same value into pairs. Take out a number from each pair and multiple the numbers up.
Example: Find the square root of 1296.
Solution: The factors of 1296 are 2, 2, 2, 2, 3, 3, 3, 3
Pair up the factors: (2,2) (2,2) (3,3) (3,3)
Take a number out from each pair and multiply them up: 2x2x3x3 = 36
The square root of 1296 is hence 36.
Exercise: Find the square root of each of the following numbers:
49, 81,144, 196, 225, 400
However, there are many numbers that do not have factors in pairs, e.g. factors of 84 are 2, 2, 3 and 7. For
such numbers, the square roots of them are not integers. So while numbers such as 4, 9, 16, 256 are
known as prefect squares, those numbers that do not have factors in pairs are not.
How to make an educated guess on the square root of a number?
Recall that 102 is 100, 202 is 400, etc. So if you are given a number less than 100, say 64, you know that
the number that you want must be less than 10 (since 102 = 100, which is greater than 64), and is between
1 and 10. You can then take the number in the middle, i.e. 5 and see if that gives what you want. Since 52
= 25 is less than 64, then you can try a number between 5 and 10, and so on. Of course, if you are very
familiar with the time table, you may identify the number much faster.
Let’s try to find the square root of 729.
a) Start with some numbers that are multiples of 10. 202 = 400 and 302=900, so the number must be
between 20 and 30.
b) Next, let’s try 25. 252 = 625 which is still too small.
c) Try a number between 25 and 30, say 27. 272=729, BINGO!
2
You can estimate the square root of a number if it is not a perfect square by using the above method.
Example: Estimate the square root of 250 to 1 decimal point.
Solution:
a) Let’s start with 102 = 100 and 202 = 400 as 250 is between the two numbers.
b) Next, try the number 15 which is in between 10 and 20. 152 = 225.
c) As 225 is smaller than 250 but is very close already, so let’s try 16. 162 = 256.
d) Now we know that the square root of 250 is between 15 and 16. Again, let’s take a number in the
middle of 15 and 16, i.e. 15.5. 15.52 = 240.25.
e) Since it is too small, let’s try 15.8. 15.82 = 249.64. BINGO!
f) So the estimated square root of 250 is 15.8
Exercise. Estimate the square roots of the following numbers to 1 decimal point.
45, 80, 360, 520
Pythagorean Theorem
In a right angle triangle, the sum of the squares of the two shorter sides, the legs, is equal to the square of
the longest side, the hypotenuse.
Let the two shorter sides of a right angle triangle be a and b, and the hypotenuse be c, then
a2 + b2 = c2
Examples:
1. A right angle triangle has legs with length 3 cm and 4 cm. Determine the length of the hypotenuse.
32 + 42 = 9 + 16 = 25
√25 = 5
The length of the hypotenuse is 5 cm.
2. If a triangle has sides equal to 5 cm, 8 cm and 12 cm. Determine if the triangle is a right angle triangle.
The two shorter legs have length 5 cm and 8 cm, so 52 + 82 = 25 + 64 = 89 which is not equal to
122 (= 144). Hence, the triangle is not a right angle triangle.
3