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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Lesson 25
Chapter 5: Indices and Standard form

Square roots and Cube Roots

Square Root
The square root of a given number is a number that, when squared, exactly equals the given
number. The square root or radical sign is
.
64  8 because 82 = 64
0.09  0.3 because (0.3)2 = 0.09
Taking a root is the inverse operation of raising to a power in the same way that subtraction is
the inverse of addition and division is the inverse of multiplication. Numbers whose square
roots are exactly whole numbers, fractions, or finite decimals, such as 64 and 0.09 shown
above, are called perfect squares. The perfect squares of the whole numbers 1 to 12 are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
You should learn these perfect squares to help you understand the ideas better and do
calculations with square roots more effectively.
Example 1
Find the square roots without the calculator:
a)
16
49
121
144
b)
Answer
To find the square root of a fraction, find the square root of the numerator and the denominator
separately:
2
a)
16
16 4
4 2 16
4

 because    2 
49
49
7
49 7
7
b)
121
121 11
112 121
 11 


because    2 
144
144
12
144 12
 12 
2
5 Indices and Standard form
Page 1
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Example 2
Find the square roots without the calculator:
a) 1.44
b)
0.0025
Answer
a)
When you square a decimal that is a perfect square the number of decimal
places
doubles. Therefore, the square root of a decimal will have half as many decimal places
as the original number. Since 1.44 has two decimal places, its square root has one
decimal place. Knowing that 122 = 144, it follows that 1.22 = 1.44 and therefore:
1.44  1.2
b)
The square root of a number less than one is greater than the number itself. This is true
because, when you square a number less than one, the result is less than the original
number. For example, 0.32 = 0.09. Since the number in the square root, 0.0025, has four
decimal places, its square root has two decimal places. Knowing that 52 = 25, it follows
that 0.052 = 0.0025 and therefore:
0.0025  0.05
Numbers that are not perfect squares, such as 2, 3, 5, 7, etc. have square roots that are infinite
decimals, and it is necessary to round off the calculator result:
2  1.414213562...  1.414
7  2.645751311...  2.646
The symbol “” means approximately equal. That is, 2 is approximately equal to 1.414, and
7 is approximately equal to 2.646. When square roots are infinite decimals, it is sometimes
easier to calculate with them in radical form instead of as decimals. The following definition of
the square root for any number x is useful in such calculations:

Definition of Square Root
x2 
 x
2
x
(3.1)
The definition says that the operations of square and square root are inverse operations and
“cancel each other out.” When a square root and a square appear together, the radical and the
square can be eliminated. For example:
5 Indices and Standard form
Page 2
Name of Lecturer: Mr. J.Agius
Course: HVAC1
9 2  9 and
 9
2
9
2
 2
22 2
2
 
 and 
2

5
5
5
 5
Two basic rules for roots of positive numbers x and y are:

Rules for Roots
xy 
 x  y 
x

y
(3.2)
x
(3.3)
y
Rules (3.2) and (3.3) are used to simplify radicals. You can apply them working from left to
right, or from right to left. Products (or quotients) under a radical sign can be separated into
products (or quotients) of separate radicals and vice versa.
Example 3
Find the square roots without the calculator:
a) 14400
b)
324
Answer
Use rule (3.2) to find the square root of a large perfect square by factorising the number into
small perfect squares. Apply the rule from left to right, separating the radical into the product of
two radicals containing perfect squares:
144100   144  100   1210  120
481   4  81  29  18
(a)
14400 
(b)
324 
Example 4
Simplify the product a)

0.2
 1.8 
b)
0.64
0.0256
c)
28
7
Answer
Apply rule (3.2) from right to left and multiply under one radical:
a)
b)
c)

0.2
 1.8  
0.21.8 
0.64
0.64


0.0256
0.0256
28
7

0.36  0.6
0.8
40.0064

0.8
4 0.0064

0.8
0.80

5
20.08 0.16
28
 42
7
5 Indices and Standard form
Page 3
Name of Lecturer: Mr. J.Agius

Course: HVAC1
Cube Roots
A cube root is the inverse of raising to the third power. A cube root is written using the radical
sign with the index 3 in the crook of the radical sign (for a square root the index 2 is
understood):
1000  10 because 103 = 1000
0.064  0.4 because 0.43 = 0.064
3
3
The first six perfect cubes are:
3
8  2,
3
27  3 ,
3
64  4 ,
3
125  5 ,
3
216  6
It is helpful to know these perfect cubes. Rules (3.2) and (3.3) for square roots also work for
cube roots, as shown in the next example.
Example 5
Find the cube root of
3
8
125
Answer
Apply rule 3.3 for cube roots. Separate the fraction into two cube roots:
3
3
8
8
2
3

125
125 5
The definition of a cube root for a number x is similar to that for a square root:

Definition of Cube Root
3
x3 
 x
3
3
x
(3.4)
Since raising to the third power is the inverse of taking a cube root, the two operations cancel
each other. For example:
3
83  8 and
 8
3
3
8
The cube root of a number can be found on the calculator by using the cube root key.
Note also that

3
1
3
x  x . This rule is valid to any root.
Raising to Positive Powers with Powers of Ten
Raise the number to the positive power and multiply the exponents to obtain the power of ten.
(5  102)3 = 53  10(2)(3) = 125  106
5 Indices and Standard form
Page 4
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Example 6
a) (1.5  10)2
Perform the operations
b) (1.84  103)3
Answer
a)
Square the 1.5 and multiply the power of ten by 2:
(1.5  10)2 = 1.52  101(2) = 2.25  102
b)
Raise 1.84 to the third power and multiply the power of 10 by 3.
(1.84 103)3 = 1.843  103(3)  6.23  109

Taking Roots with Powers of Ten
Take the root of the number and divide the root index into the power of ten.
6
2
81  10  81  10  9  10 3
6
Example 7
Simplify
a)
2.25  10 4
b)
27  10 9
3
Answer
a)
Take the square root of 2.25 and divide the exponent 4 by the index 2:
4
2
2.25  10  2.25  10  1.5  10 2
4
b)
Take the cube root of 27 and divide the exponent 9 by the index 3.
3
9
3
27  10  27  10  3  10 3
9
3
Example 8
Calculate and give the answer in power of ten, and in ordinary notation:
9  10 6
2  10 
3 2
Answer
Apply the order of operations, and do the powers and roots first. Find the square root of the
numerator and raise the number to the power in the denominator.
9  10 6
2  10 
3 2
Then do the division:
5 Indices and Standard form
6
9  10 2 3  10 3
 2

2  10 32  4  10 6
3  10 3 3
  10 36  0.75  10 3  0.00075
4  10 6 4
Page 5
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Square roots and Cube roots
Q1
Work out the following without using the calculator. Do each exercise by hand and give the
answers in decimal or fraction form.
a)
16
b)
e)
16
36
f)
i)
0.01
j)
121
4
81
2.25
m)
3
27
n)
3
125
q)
3
0.001
r)
3
0.008
Q2
c)
25
4
d)
9
100
g)
0.64
h)
0.36
k)
0.0081
l)
0.0144
1
8
p)
o)
3
3
64
1000
Simplify each expression by applying the rules for radicals. Do each exercise by hand and give
the answers in decimal or fraction form.
a)
 10 
2
b)
0.12
c)
784
d)
576
e)
1.96
f)
4.84
g)
8 18
h)
12 27
i)
0.4 0.9
j)
0.02 0.08
k)
n)
0 .2
0 .8
m)
Q3
a)
e)
Q4
0.09
9
12
3
(4  103)3
(0.8  105)2
b) (6  102)2
f) (0.1  10)2
c) (2  102)4
g) (1  103)(3  104)2
(3  104)3
(4  10)(2  103)3
d)
h)
Work out the following without the use of a calculator and express the answer in terms of
power of ten.
16  10 6
b)
25 10 4
c)
e)
0.25  10 4
f)
0.64  10 6
g)
3
k)
m)
75
Work out the following without the use of a calculator and express the answer in terms of
power of ten.
a)
i)
3
l)
 9 10  4 10 
4
2
36  10 2
2  10 
3 2
5 Indices and Standard form
j)
n)
 110  25 10 
6
64  10 4
5  10 
2 2
2
144  10 2
d)
8 10 9
h)
9  10 8
15  10 3
l)
3  10 
3 2
o)
144  10 2
49  1012
3
27  10 3
16  10 2
8  10 2
6  10 
2 2
p)
81  10 4
Page 6
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Applied Problems
Q4
In the following problems try to solve each by hand. Check with the calculator.
a)
The area of a square plot of land is A = 4900 ft2. Find the side of the square s 
b)
The area A of a square circuit board is 10,000 mm2 (square millimetres). How long is the side
s  A of the circuit board in mm and cm (centimetres)?
c)
Find the hypotenuse of a right triangle given by: c  1.5 2  2.0 2
d)
The radius of a sphere is given approximately by: r  3
A.
V
where V = volume. Find r in ft,
4.2
when V = 0.0042 ft3.
Applications to Electronics
Q5
Work out with the use of a calculator
a)
The impedance of an alternating current (ac) circuit is given by:
Z  16 2  12 2
Impedance is measured in ohms like resistance. Find the value of Z in ohms ().
b)
As in problem (a), find the impedance Z in ohms of an ac circuit given by: Z  15 2  20 2
c)
Find the voltage V in volts (V) in a circuit given by: V 
12030
d)
Find the current I in amps (A) in a circuit given by: I 
4
10000
e)
Find the current I in amps (A) in a circuit given by: I 
10
150  100
f)
Find the voltage V in volts (V) in a circuit given by: V 
16100  44
3.2  10 
P
3 2
g)
Find the power P in watts (W) dissipated as heat in a resistor given by:
h)
Find the power P in watts (W) dissipated as heat in a resistor given by: P = (0.12)2(0.5  103)
i)
Find the current I in amps (A) in a series circuit given by:
I
1.6  10 6
24
5.7  10  3.9  10 3

3
 

Change to ordinary notation before calculating.
j)
Find
V
the
voltage
166.110
3
drop
 3.9  10
5 Indices and Standard form
3

V
in
volts
across
two
resistances
given
by:
Page 7