Maths Learning Service: Revision
Mathematics IA
Index Laws
Mathematics IMA
Intro. to Fin. Maths I
Index laws are the rules for simplifying expressions involving powers of the same base number.
am × an = am+n
(am )n = amn
am
= am−n
an
1
a−m = m
a
First Index Law
Second Index Law
Third Index Law
a0 = 1
1
an =
√
n
a
Examples: Simplify the following expressions, leaving only positive indices in the answer.
(a)
3
36 24
34
36
= 4 × 24
3
(b)
−5
2
3 ×3
= 3−3
1
33
1
=
27
= 32 24
=
9 (x2 )
(c)
3xy 2
9 x6
1
= ×
× 2
3
x
y
1
= 3 × x5 × 2
y
5
3x
= 2
y
(d)
√
a−1 a
1
= a−1 a 2
1
= a− 2
=
1
1
or √
a
a
1
2
Notes: (1) More involved fractional powers can be dealt with by noting that
m
1
a n = (a n )m by the Second Index Law. For example,
√
2
1
3
(27) 3 = (27 3 )2 = ( 27)2 = (3)2 = 9.
(2)
Watch out for powers of negative numbers. For example,
(−2)3 = −8 and (−2)4 = 16, so (−x)5 = −x5 and (−x)6 = x6 .
Index Laws
(3)
2007 Maths IA, IMA & Intro. Fin. Maths I Revision/2
In general (ab)n = an bn . For example,
(3x2 y)3 = 33 (x2 )3 y 3 = 27x6 y 3 .
Exercises
1. Simplify the following expressions, leaving only positive indices in the answer.
−2
(a)
42 × 4−3
(b)
32 (22 )
23
(c)
x5 x8
(d)
(y 4 )6
(e)
(−3)3
(f)
(4ab2 c)3
(g)
x2 z −3 × (xz 2 )2
(h) 2n × (2−n )3 × 22n
(i)
3m × 27m × 9−m
(j)
(a 2 × a)5
(l)
(−a4 b)3 (ab)5
−a8 b8
(m)
x−1 y 4
x−5 y −3
(o)
√
x3x
(r)
(3a)−1 × 3a−1
(u)
1
2
(p)
(a ×
(s)
3
5
32
√
(k)
(−2ab)2
2b
10a3 b−2
5a−1 b2
(n)
!−1
1
2
a)
(q)
2x 2 x
x2
(t)
4
25
3
2
1
43
1
23
√
Terms involving the “ ” symbol are known as a radicals or surds.
√
√
√
√
√
Notes: (1)
a + b 6= a + b . For example 144 + 25 = 169 = 13
√
√
but 144 + 25 = 12 + 5 = 17.
√
√
√
(2) Similarly, a − b 6= a − b.
√
√
√
√ √
√
√
(3)
ab = a × b . For example 4 × 9 = 36 = 6 and 4× 9 = 2×3 = 6.
s
√
√
r
16 √
16
4
a
a
(4)
= 4 = 2 and √ = = 2.
= √ . For example
4
2
b
4
b
These techniques can be used to simplify radicals. For example
√
√
√
9× 2=3 2.
√
√
√
75 = 25 × 3 = 5 3 .
18 =
√
9×2=
√
When asked to simplify radical expressions involving fractions, you are required to produce
a single fraction (as in ordinary algebra) with no radicals in the denominator. For example
Index Laws
2007 Maths IA, IMA & Intro. Fin. Maths I Revision/3
√
√
√
3
3
3
2
2
√ +√ = √ ×√ +√
2
6
3
2
6
3
2
= √ +√
6
6
5
= √
6
√
5
6
= √ ×√
6
6
√
5 6
=
6
Exercises (continued)
2. Simplify the following expressions
√
√
(a)
50
(b)
72
√
√
3
2
1
1
(e) √ + √
(d) √ − √
2
3
10
5
(c)
(f)
√
12 +
√
27
√
2 3
1
√ −√
15
3
Index Laws
2007 Maths IA, IMA & Intro. Fin. Maths I Revision/4
Answers to Exercises
1.
(a)
1
4
(b)
9
9
=
7
2
127
(c) x13
(d) y 24
(e) −27
(j) a15/2
(f) 64a3 b6 c3
(g) x4 z
(h) 1
(i) 32m
(k) 2a2 b
(l) a9
(m) x4 y 7
(n)
(p) a5
(q) 2x−1/2
(r) a−2
(s) 8
√
(b) 6 2
√
√
5 3−6 5
(f)
15
√
(c) 5 3
1
2
a−4 b4
(u) 2
√
2. (a) 5 2
√
5 6
(e)
6
√
√
2 5 − 10
(d)
10
(o) x4/3
(t)
8
125