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UNIT 3 VOCABULARY: POWERS AND ROOTS
1.1. Powers with natural base
An exponent is a short way of writing the same number multiplied by itself
many times. Exponents can also be called indices (singular index).
In general, for any real number a and natural number n, we can write a
an .
multiplied by itself n times as
REMEMBER: READING POWERS!
A power can be read in many ways in English. For example,
• The fifth power of six.
• Six powered to five.
•
65 can be read as:
The most common one: six to the power of five.
Remeber that there are two especial cases: squares and cubes (powers of two and three). For
example:
32 is read three squared.
53 is read five cubed.
Exercise. Write with words these expressions and work out the exact value:
a)
25
b)
106
c)
73
d) 4 2
e) 34
f)
1.2. Laws of exponents
There are several laws we can use to make working with exponential numbers easier.
Some of these laws were explained last year:
LAW
EXAMPLE
x 0=1
7 0 =1
a =a
51=5
a m⋅an =a m+n
x 2⋅x 3=x 2+3=x 5
1
am
a
26
=a m−n
n
2
or
a m a n=am−n
n
n
n
(a⋅b) =a b
2
38
3
7
=26−2=24
=38−7=3
(2⋅3)4 =24⋅34
83
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n
()
a
b
=
a
n
b
n
5
Exercises.
1. Simplify as far as possible:
a) a · a · a
c) 10 · k · j · k · j
b) x · x · y · y · y d) 5 · y · 4 · y
2. Simplify:
4y 2⋅5y
a)
b)
7m 2⋅3n
c)
d)
3
()
2
6a⋅5a⋅2a
2a3⋅3a 2
=
23
5
3
=
8
125
e) a · b · c · 5
f) a · b · b · a · 3
g) 3 · x · x · 4 · x
h) 2 · y · x · 4
e)
f)
g)
h)
3a⋅5a⋅4
3p⋅2p 2⋅4p 3
3. Simplify (don't forget to cancel coefficients):
10x 4
15a 5
4b 3
a)
b)
c)
3
2
4
5x
3a
12b
d)
5k 2
20k
5
4a2⋅3b 5
a b a 2b 2
e)
18x5 y 4
4
27x y
5
1.3. Powers with negative base or negative exponents
The sign of a power is positive, unless the base is negative and the exponent is an odd number.
Base
Exponent
Sign of the result
Example
+
Odd or even
+
23=8
24=16
-
Even
+
(−2)4 =(−2)⋅(−2)⋅(−2)⋅(−2)=16
-
Odd
-
(−2)3=(−2)⋅(−2)⋅(−2)=−8
BE CAREFUL WITH BRACKETS!!!!
Brackets are very important in Maths, and you have to be very careful with them. Have a
look at the following examples:
With ( )
(−2)2=(−2)⋅(−2)=4
Without ( )
−22=−2⋅2=−4
With ( )
( ab )2=ab ⋅ab =a 2 b 2
Without ( )
ab 2= a⋅b⋅b =ab 2
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Exercise. Work out these powers:
2
a)
5
b)
2
c)
4
d)
2
()
1
e)
3
3
f)
(−3)
4
−(−3)
2
(−2)
2
g)
(1+5)
h)
2 +3
2
i)
2
j)
4
4
2
()
()()
( )( )
1
k)
2
1
2
2
2
1
÷
2
3
2
⋅
1
3
In general, a negative exponent can be calculated following this rule:
For example:
4−2=
1
4
2
=
1
16
10−3=
1
10
3
=
1
1000
(−2)−3=
1
3
(−2)
=−
1
8
1
5
−2
=5 2=25
Exercises.
1. Work out:
a)
b)
3−4
c)
−2
(−4)
d)
10−4
e)
1
f)
−3
−3
2. Write using negative exponents:
1
1
a)
b)
c)
2
2
10
x
g)
(−3)−3
h)
7
−2
1
10
4
a 10
d)
3
−
e)
2.1. Square root of a number
A square root of a number is a value that can be multiplied by itself to
give the original number. A square root is the opposite of a square:
For example, a square root of 9 is 3, because when 3 is multiplied by
itself you get 9.
This is the special symbol that means "square root". It is
also called "radical" symbol.
It is very easy to read a square root:
√ 49=7
is read "the square root of 49 is 7"
In general:
√ a=b
because
2
b =a
−4
()
( )
2
( )
f)
−
−
2
5
−2
2
5
52
a
4
−3
i)
f)
1
5
3−2
4
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In some cases, there can be more than one root (or less than one!):
√a
Number of roots
Example
a >0
Two opposite roots
√ 81=±9
2
9 =81
a =0
One root
a <0
No real roots
Exercises.
1. Given y², find the value of y:
a) y² = 81
b) y² =16
2. Fill
a)
b)
c)
d)
e)
c) y² = 100
because
and (−9)2=81
√ 0=0
√−4
d) y² = 4
e)
has no real roots
y² = 36
in the gaps with the proper word:
The ______ root of 36 is 6 or -6 because ______ and ______ are equal to 36.
The square root of 64 is ___ or ___ because 24 and _____ are equal to 64.
The square ____ of ____ is ____ or ____ because 122=144 and ______ .
The square root of 10000 is _____ or ____ because ______ and ______ .
The square root of -121 can't be calculated because the ______ is a ______ number.
3. Write the different parts of the expression
Radicand _____
√ 169=13
Radical _____
Root _____
2.2. Why are square roots useful?
If you know the area of a square, you can use the square root to find the length of the side of
the square. Look:
2.3. Calculating squares roots
The perfect squares are the squares of the whole numbers:
Perfect
Squares
1
2
3
4
5
6
7
8
9
10
11
12
13
...
1
4
9
16
25
36
49
64
81
100
121
144
169
...
It is easy to work out the square root of a perfect square, but it is really hard to work out other
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square roots.
√ 25=5
√ 169=13
For example, what is
√ 10
√ 100=10
√ 900=30
?
Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.
3<√ 10<4
Let's try 3.5: 3.5 × 3.5 = 12.25
•
•
•
Let's try 3.2: 3.2 × 3.2 = 10.24
Let's try 3.1: 3.1 × 3.1 = 9.61
Getting closer to 10, but it will take a long time to get a good answer!
At this point, I get out my calculator and it says:
√ 10 = 3.1622776601683793319988935444327 ...
But the digits just go on and on, without any pattern.
So even the calculator's answer is only an approximation !
Remember: if a square root is not exact, it is then an irrational number!
Exercise. Find pairs of consecutive numbers that bound:
a) __ <
√ 50
< __
b) __ <
√ 150
< __
c) __ <
√ 115
< __
d) __ <
√2
< __
2.4. Properties
Exercise. Write an example for every property of square roots. After that your teacher will explain
how it works.
Name
Property
Example
Product of square roots
√ a⋅√ b= √ a⋅b
√a = a
√b b
√
Division of square roots
Exercise. Simplify:
a)
√ 16 x
2
b)
√ 25 a
2
c)
√ 300
√3
d)
√ 100 m
2
e)
√ 64 a
2
b
2
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3.4. Order of operations
•
•
•
•
Do all operations in brackets and square brackets first.
Work out the exponents and square roots.
Then do multiplications and divisions in the order they appear (from left to right).
Finally, do additions and subtractions in the order they appear.
Easy way to remember them!
Brackets
Exponents
Divisions
Multiplications
BEDMAS
Additions
Subtractions
For example:
•
•
[(− 2)5 − (− 3)3]2 = [− 32 − (− 27)] = (−32 + 27)2 = (−5)2 = 25
2
(56−7 )⋅√ 25=(56−49)⋅5=7⋅5=35
4.1. Cube Root
A cube root of a number a special value that when cubed gives
the original number. A cube root is the opposite of a cube:
For example, a cube root of 27 is 3, because when 3³ is 27.
This is the special symbol that means "cube root".
It is very easy to read a cube root:
3
√ 125=5
is read "the cube root of 125 is 2"
In general:
3
√ a=b
because
3
b =a
EVERY NUMBER HAS A SINGLE CUBE ROOT!
Negative numbers also have a cube root. Look at the example:
3
√64=4
because
3
4 =64
BUT
3
√−64=−4
because
3
(−4) =−64
Exercise. Work out:
a)
3
√ 125
b)
3
√−1000
c)
3
√−8
d)
3
√−1
e)
3
√ 8000
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4.2. Why are cube roots useful?
If you know the volume of a cube, you can use the cube root to find the length of the edge of the
cube. Look:
4.3. Calculating cube roots
The perfect cubes are the cubes of the whole numbers:
Perfect
Cubes
1
2
3
4
5
6
7
8
9
1
8
27
64
125
216
343
512
729
10
11
12
13
1000 1331 1728 2197
...
...
It is easy to work out the cube root of a perfect cube, but it is really hard to work out other
cube roots.
3
√ 169=13
√−216=−6
For example, what is
3
√ 30
3
√ 100=10
√−8=−2
?
Well, 3 x 3 x 3 = 27 and 4 x 4 x 4 = 64, so we can guess the answer is between 3 and 4.
3
3<√ 30<4
•
Let's try 3.5: 3.5 × 3.5 × 3.5 = 42.875
•
Let's try 3.2: 3.2 × 3.2 × 3.2 = 32.768
•
Let's try 3.1: 3.1 × 3.1 × 3.1 = 29.791
We are getting closer, but very slowly.
At this point, I get out my calculator and it says:
3.1072325059538588668776624275224 ...
But the digits just go on and on, without any pattern.
So even the calculator's answer is only an approximation !
If a square root is not exact, it is then an irrational number!
Exercise. Find pairs of consecutive numbers that bound:
a) __ <
3
√ 80
< __
b) __ <
3
√ 150
< __
c) __ <
3
√5
< __
d) __ <
3
√ 300
< __
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4.4. Properties
Exercise. Write an example for every property of cube roots. After that your teacher will explain
how it works.
Name
Property
Product of cube roots
3
3
Example
3
√ a⋅√ b= √ a⋅b
√a = a
√b b
3
Division of cube roots
√
3
3
Exercise. Simplify:
3
a)
3
√ 27 a
3
b)
3
√ 64 a
3
b
3
c)
√ 270
√ 10
3
d)
3
√ 1000 x
3
e)
3
√a
6