Name of Lecturer: Mr. J.Agius
Course: FCES
LESSON 5
Squares, Cubes, Square roots & Estimation
5.1 Squares
We obtain the square of a number when we multiply the number by itself.
Example 1
Find the square of:
a) 4
b) 0.02
Answer
a) 42 = 4  4 = 16
b) 0.022 = 0.02  0.02 = 0.0004
Note: Pay attention how you multiply decimal numbers.
Exercise 1
Find the squares of the numbers:
1)
300
2)
0.004
3)
2000
4)
0.2
Example 2
Write 32 correct to 1 s.f. and use this to give a rough estimate of the square of 32
Answer
3|2  30
322  302 = 30  30 = 900
Exercise 2
In the following questions, give each number correct to 1 s.f. Then use this to give a
rough estimate of the square of the number.
1)
99
2)
0.081
3)
Learning Outcome 1 – Numerical Calculations
249
4)
37.2
Page 36
Name of Lecturer: Mr. J.Agius
Course: FCES
5.2 Square Roots
The square root of a number is the number which, when multiplied by itself, gives the
original number,
e.g. since 42 = 16, the square root of 16 is 4.
The symbol for the positive square root is
16  4
So
Example 3
Find the square roots of
a)
b)
c)
Answer:
a)
In fact 5  5 = 25
b)
In fact 700  700 = 490000
Note: In order to work such number, you have to pair the number from the decimal
point. Then see what the last number near the square root is and calculate its
square root. Afterwards, from each pair of zeros, a zero is written instead.
i.e.
(two pair of zeros in the square root)
c)
Note. In order to work this out, we can follow the same reasoning as above. I.e. by
pairing from the decimal point.
i.e.
(one pair of zeros)
Note: The zero before the decimal point is always written.
Learning Outcome 1 – Numerical Calculations
Page 37
Name of Lecturer: Mr. J.Agius
Course: FCES
Exercise 3
Find the square root of:
1)
81
2)
40000
3)
0.09
4)
0.000016
5.3 Rough Estimates of Square Roots
So far, we have been able to find exact square roots of the numbers we have been given.
Most numbers, however, do not have exact square roots; 23 , for example, lies between
4 and 5 because 4  4 = 16, and 5  5 = 25.
23 , if given as a decimal, will start with 4.
23  4.   
i.e.
Example 4
Find the first significant figure of the square root of 30.
Answer
(Check: 5  5 = 25 and 6  6 = 36. So the first significant figure should be 5 since
the square number of 5 is less than 30)
Exercise 4
Find the first significant figure of the square roots of the following numbers:
1)
17
2)
4.6
3)
Learning Outcome 1 – Numerical Calculations
90
4)
15
Page 38
Name of Lecturer: Mr. J.Agius
Course: FCES
Exercise 5
1
Write how many pegs in each board
2
3
4
5
6
7
8
Write how many pegs would fit on a:
9
20 × 20
board
10
30 × 30
board
11
40 × 40
board
Learning Outcome 1 – Numerical Calculations
12
50× 50
board
13
15 × 15
board
Page 39
Name of Lecturer: Mr. J.Agius
Course: FCES
Exercise 6
Estimate the square of each number and then use your
calculator to find the exact square numbers:
1
21
2
32
3
17
4
44
5
19
6
26
7
52
8
13
9
16
10
29
11
25
12
121
Note: A student is expected to know the first 10 square
numbers by heart
Complete the following tables
Number
1
2
3
4
5
6
7
8
9
10
11
12
Square
Number
20
30
40
50
60
70
80
90
100
200
300
400
Square
Number
Square
Learning Outcome 1 – Numerical Calculations
Page 40
Name of Lecturer: Mr. J.Agius
Course: FCES
Exercise 7
Estimate the square of each number and then use your
calculator to find the exact square numbers:
1
41
2
9.5
3
21
4
17
5
89
6
6.25
7
15
8
25
9
101
10
500
11
69
12
55
Write the missing numbers
without the use of the
calculator
122 = 144
152 = 225
502 = 2500
1
15 × 16 =
2
13 × 12 =
3
49 × 50 =
4
12 × 11 =
5
14 × 15 =
6
50 × 51 =
Learning Outcome 1 – Numerical Calculations
Page 41
Name of Lecturer: Mr. J.Agius
Course: FCES
Exercise 8
Write the lengths of the sides
for each square picture.
1
2
Area 2601 cm2
4
3
Area 484 cm2
5
Area 20.25 m2
7
Area 2025 cm2
6
Area 1.44 m2
8
Area 196 cm2
Learning Outcome 1 – Numerical Calculations
Area 12.25 cm2
9
Area 289 m2
Area 0.25 m2
Page 42
Name of Lecturer: Mr. J.Agius
Course: FCES
5.4 Cubes and Cube Roots
Up till now, we have learned that to find the square number, we have to multiply the
same number twice. To find the cube of a number we have to multiply the same number
three times.
Example 5:
The cube of 2 (i.e. 23) = 2  2  2 = 8
Note: A student is expected to know the first 5 cube numbers
by heart
Exercise 9:
1
Number
Cube
2
3
4
5
As square roots are the opposite of square numbers, cube roots are the opposite of cube
numbers
Example 6:
The cube root of 64 ( 3 64 ) = 4
(because 4  4  4 = 64)
Exercise 10:
Work out the cube roots of
a) 125
b) 8
Learning Outcome 1 – Numerical Calculations
c) 27
d) 1
Page 43
Name of Lecturer: Mr. J.Agius
Course: FCES
5.5 Using the Calculator
Square Numbers
To find the square number with the calculator you have to use the button
x2
.
First you have to press the number given and then press the square button and the answer
will appear automatically.
Exercise 11:
Find the squares of the following numbers with the use of the
calculator.
a) 21 Ans. _______
b) 33 Ans. _______
c) 65 Ans. _______
d) 24 Ans. _______
e) 46 Ans. _______
f) 123 Ans. _______
g) 44.1 Ans. _______
h) 3.2 Ans. _______
i) 63.2 Ans. _______
Cube Numbers and Higher Powers
To find the cube numbers with the calculator you have to use the button
Or
^
xy
(This button means to the power of. This button can be used for
higher index numbers)
To use this button, first press the whole number on your calculator, then press this button
and hence press the index number you have. Don’t forget that this button can be used for
any index.
Example 8:
Find the cube number of 14.
Answer
143 = 14
Exercise 12:
Find the cube number of:
^
3 = 2744
a) 23 Ans: _____
b) 12 Ans: _____
c) 36 Ans: _____
d) 1.4 Ans: _____
e) 8.2 Ans: _____
f) 6
b) 28
Ans: _____
Find the value of:
a) 65
Ans: _____
Learning Outcome 1 – Numerical Calculations
Ans: _____
c) 46
Ans: _____
Page 44
Name of Lecturer: Mr. J.Agius
Course: FCES
Square Roots
To find the square root of a number with the calculator you have to use the button

So in order to work out the square root of a number first press the square root button, then
press the number and press the equal sign. Note some calculators work the other way
round. So check carefully how you calculator works.
Example 9: Find the square root of 56.
Answer
56  7.48
Exercise 13:
to 3 significant numbers
Find the square root of each number to 3 significant figures.
a) 43 Ans: _____
b) 89 Ans: _____
c) 432 Ans: _____
d) 83 Ans: _____
e) 754 Ans: _____
f) 1234321
g) 129.6
Ans: _____
h) 3969
Ans: _____
Ans: _____
Cube Roots and Higher Powers
To find the cube root of a number with the calculator you have to use the button
So in order to work out the cube root of a number first press the index number, then press
the cube root button, then press the whole number given and press the equal sign. Note
some calculators work the other way round. So check carefully how you calculator
works. Don’t forget that this button can be used for higher order index.
Example 10: Find the cube root of 216.
Answer
3
216  3
216 = 6
Exercise 14: Find the cube root of each number to 3 significant figures.
a) 166375
Ans: _____
b) 17576
Ans: _____
c) 1728
Ans: _____
d) 512
Ans: _____
e) 704969
Ans: _____
f) 54872
Ans: _____
Find the value of:
a)
8
256
Ans: _____
Learning Outcome 1 – Numerical Calculations
b)
5
243
Ans: _____
Page 45
Name of Lecturer: Mr. J.Agius
Course: FCES
Mixed Exercises
In these following exercises you can use the calculator.
Find the value of:
a) 46
Ans: _____
b) 32
Ans: _____
c) 93
Ans: _____
Ans: _____
d) 196
Ans: _____
e) 1024
Ans: _____
f) 33
Ans: _____
g) 64
Ans: _____
h) 92
Ans: _____
i) 100
Ans: _____
j) 53
Ans: _____
k) 182
Ans: _____
9
Ans: _____
m) 4 1296
Ans: _____
n)
3
Ans: _____
o) 120
Ans: _____
p) 153
Ans: _____
q) 113
Ans: _____
r) 114
Ans: _____
s) 115
Ans: _____
t)
5
Ans: _____
d)
l)
81
512
Learning Outcome 1 – Numerical Calculations
2197
248832
Page 46