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Index Law s
1. Index notation
Rather than write 2 · 2 · 2 · 2 · 2, we write such a product as 25 .
25 reads “two to the power of five”, “two raised to five” or “two with index five”.
If n is a positive integer, then an is the product of n factors of a
i.e., an = |a · a · a{z· a . . . a}
n factors
a is the base.
n is the power, index or exponent.
Other examples:
52 reads “five to the power of two”, “five raised to two” or “five squared”.
73 reads “seven to the power of three”, “seven raised to three” or “seven cubed”.
Exercises - Set A
1. What is the last digit of 3100 ?
(Hint: Consider 31 , 32 , 33 , 34 , 35 , 36 . . . and look for a pattern.)
2. What is the last digit of 7200 ?
2. Negative bases
So far we have only considered positive bases raised to a power.
We will now look at negative bases. Consider the statements below:
(−1)1 = −1
2
(−1) = (−1) · (−1) = 1
(−1)3 = (−1) · (−1) · (−1) = −1
(−1)4 = (−1) · (−1) · (−1) · (−1) = 1
(−2)1 = −2
(−2)2 = (−2) · (−2) = 4
(−2)3 = (−2) · (−2) · (−2) = −8
(−2)4 = (−2) · (−2) · (−2) · (−2) = 16
In he pattern above it can be seen that:
A negative base raised to an odd power is negative; whereas
a negative base raised to an even power is positive.
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Index law s - 3 o E SO
2
Be careful:
to a power.
(−2)4 = 16, but −24 = −16, so always use brackets when raising a negative base
Exercises - Set B
1. Simplify:
a) (−1)17
b) −54
c) (−2)5
a) 2, 86
2. Find using your calculator:
b) (−5)5
d) −(−3)2
c) −94
d) (−1, 14)23
3. Find using your calculator, correct to 3 decimal places:
a) (2, 6 + 3, 7)4
3
3, 2 + 1, 92
d)
1, 47
b) 8, 63 − 4, 23
4
0, 52
e)
0, 09·, 14
c) 12, 4 · 10.74
a)
648
3, 624
3. Index laws
Recall the following index laws where the indices m and n are positive integers:
am · an = am+n
To multiply numbers with the same base, keep the base and add
the indices.
am
= am−n
an
To divide numbers with the same base, keep the base and subtract the indices.
(am )n = am·n
When raising a power to a power, keep the base and multiply
the indices.
(a · b)n = an · bn
a an
= n
b
b
The power of a product is the product of the powers.
The power of a quotient is the quotient of the powers.
Exercises - Set C
1. Simplify using the index laws (leave your answer in index form):
a) 22 · 24
b) 114 · 11
c) 1315 · 136
d) 172 · 175
e)
f)
g) 149 : 144
h) 98 : 92
25
22
56
5
i) (22 )4
j) (104 )2
k) (93 )7
l) (74 )5
m) 63 · 53
n) 84 : 24
o) 57 · 37
p) (63 )2 · 96
q) (83 )3 · 82
r) (102 )5 : 210
s) (62 )3 · 6
t) (73 )4 : (72 )4
4. Zero and negative indices
Look at this example:
23
8
= =1
23
8
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Index law s - 3 o E SO
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23
= 23−3 = 20 .
23
But, applying the second index law:
20 = 1.
So, we deduce that:
a0 = 1, for all a 6= 0.
In general:
72
7·7
1
1
=
=
= 3
75
7·7·7·7·7
7·7·7
7
72
But, applying the second index law:
= 72−5 = 7−3 .
75
1
So, we deduce that:
7−3 = 3 .
7
Now, look at this example:
In general, if a is any non-zero number, and n is an integer, then:
• a−n =
1
an
(i.e., an and a−n are reciprocals of one another).
• In particular, a−1 =
1
a
and
a −n
b
=
n
b
.
a
Example 1
Simplify, giving answers in simplest rational form:
−2
3
−2
a) 5
b)
c) 80 − 8−1
5
a) 5−2 =
1
1
=
52
25
b)
−2 2
3
5
52
25
=
= 2 =
5
3
3
9
c) 80 − 8−1 = 1 −
1
7
=
8
8
Exercises - Set D
1. Simplify, giving answers in simplest rational form:
a) 5 − 70
b) 60 − 20
c) (6 − 2)0
d) 4−1
e) 3−2
−1
1
i)
3
f) 3−3
−1
2
j)
5
h) 2−4
m) 20 + 21 + 2−1
n) 1 21
g) 10−5
−2
3
k)
4
−2
1
o)
+ 2−1
3
−3
l) 50 − 5−1
p)
−2 4
4
1
−
3
2
2. Write as powers of 2, 3 or 5:
a) 8
f)
1
125
b)
1
8
g) 32
c) 9
h)
1
32
d)
1
9
i) 81
e) 125
j)
1
81
D pto. M atemáticas. IES Jovellanos. 2012
Index law s - 3 o E SO
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Exercises - Set E
1. Simplify:
a) 73 · 72
b) 54 · 53
c) a7 · a2
d) a4 · a
e) b8 · b5
f) a3 · an
g) b7 · bm
h) m4 · m2 · m3
k) 77 : 74
l)
59
52
b10
m) 7
b
1113
119
p5
n) m
p
i)
j)
o)
a6
a2
ya
y5
p) b2x : b
q) (32 )4
r) (53 )5
s) (24 )7
t) (a5 )2
u) (p4 )5
v) (b5 )n
w) (xy )3
x) (a2x )5
2. Express in simplest form with a prime number base:
a) 8
b) 25
c) 27
d) 43
e) 92
f) 3a · 9
g) 5t : 5
h) 3n · 9n
k) (54 )x−1
l) 2x · 22−x
3x+1
3x−1
4y
n) x
8
16
2x
2y
m) x
4
i)
j)
o)
3x+1
31−x
p)
2t · 4t
8t−1
3. Remove the brackets of:
a) (a · b)3
b) (a · c)4
c) (b · c)5
d) (a · b · c)3
e) (2a)4
f) (5b)2
g) (3n)4
i) (4abc)3
j)
h) (2bc)3
5
2c
l)
d
a 3
k)
b
m 4
n
4. Express the following in simplest form, without brackets:
a) (2b4 )3
3 4
m
e)
2n2
b) (5a4 b)2
4 2
4a
f)
b2
c) (−6b2 )2
3
−2a2
g)
b2
d) (−2a)3
2
−3p2
h)
q3
5. Rational indices
Look at this example:
1
1
1
1
Since 3 2 · 3 2 = 3 2 + 2 = 31 = 3, and
√
1
3 2 = 3.
√ √
3 · 3 = 3 also, then, by direct comparison:
√ √ √
1
1
1
Likewise, 2 3 · 2 3 · 2 3 = 21 = 2, compared with 3 2 · 3 2 · 3 2 = 2, suggests:
√
1
2 3 = 3 2.
√
√
1
In general:
an = n a
( n a reads “the nth root of a”)
m
1
Also: a n = (am ) n =
√
n
am
D pto. M atemáticas. IES Jovellanos. 2012
Index law s - 3 o E SO
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Example 2
Write as a single power of 2:
√
1
a) 3 2
b) √
2
a)
√
3
c)
√
5
4
1
1
1
b) √ = 1 = 2− 2
2
22
1
2 = 23
c)
√
√
2
5
5
4 = 22 = 2 5
Exercises - Set F
1. Write as a single power of 2:
√
√
1
a) 5 2
b) √
c) 2 2
5
2
√
√
4
f) 2 · 3 2
g) √
h) ( 2)3
2
2. Write as a single power of 3:
√
√
1
a) 3 3
b) √
c) 4 3
3
3
√
d) 4 2
1
i) √
3
16
√
d) 3 3
1
e) √
3
2
1
j) √
8
1
e) √
9 3
3. Write the following in the form ax where a is a prime number and x is rational:
√
√
√
√
√
a) 3 7
b) 4 27
c) 5 16
d) 3 32
e) 7 49
1
f) √
3
7
1
g) √
4
27
1
h) √
5
16
1
i) √
3
32
1
j) √
7
49
✁✃✁✃✁✃✁✃✁✃✁✃✁✃
D pto. M atemáticas. IES Jovellanos. 2012