Basic algebra 65
(2)am
an=am−n For example,c^5
c^2=c^5 −^2 =c^3(3) (am)n=amn For example,(
d^2) 3
=d^2 ×^3 =d^6(4) am
n=n√
am For example,x4(^3) =^3
√
x^4
(5) a−n=
1
an
For example, 3−^2 =
1
32
1
9
(6) a^0 = 1 For example, 17^0 = 1
Here are some worked examples to demonstrate these
laws of indices.
Problem 17. Simplifya^2 b^3 c×ab^2 c^5
a^2 b^3 c×ab^2 c^5 =a^2 ×b^3 ×c×a×b^2 ×c^5
=a^2 ×b^3 ×c^1 ×a^1 ×b^2 ×c^5
Grouping together like terms gives
a^2 ×a^1 ×b^3 ×b^2 ×c^1 ×c^5
Using law (1) of indices gives
a^2 +^1 ×b^3 +^2 ×c^1 +^5 =a^3 ×b^5 ×c^6
i.e. a^2 b^3 c×ab^2 c^5 =a^3 b^5 c^6
Problem 18. Simplifya
1
(^3) b
3
(^2) c−^2 ×a
1
(^6) b
1
(^2) c
Usinglaw(1)ofindices,
a
1
(^3) b
3
(^2) c−^2 ×a
1
(^6) b
1
(^2) c=a
1
3 +
1
(^6) ×b
3
2 +
1
(^2) ×c−^2 +^1
=a
1
(^2) b^2 c−^1
Problem 19. Simplify
x^5 y^2 z
x^2 yz^3
x^5 y^2 z
x^2 yz^3
x^5 ×y^2 ×z
x^2 ×y×z^3
x^5
x^2
×
y^2
y^1
×
z
z^3
=x^5 −^2 ×y^2 −^1 ×z^1 −^3 by law (2) of indices
=x^3 ×y^1 ×z−^2
=x^3 yz−^2 or
x^3 y
z^2
Problem 20. Simplify
a^3 b^2 c^4
abc−^2
and evaluate when
a= 3 ,b=
1
4
andc= 2
Usinglaw(2)ofindices,
a^3
a
=a^3 −^1 =a^2 ,
b^2
b
=b^2 −^1 =band
c^4
c−^2
=c^4 −−^2 =c^6
Thus,
a^3 b^2 c^4
abc−^2
=a^2 bc^6
Whena= 3 ,b=
1
4
andc=2,
a^2 bc^6 =( 3 )^2
(
1
4
)
( 2 )^6 =( 9 )
(
1
4
)
( 64 )= 144
Problem 21. Simplify(p^3 )^2 (q^2 )^4
Using law (3) of indices gives
(p^3 )^2 (q^2 )^4 =p^3 ×^2 ×q^2 ×^4
=p^6 q^8
Problem 22. Simplify
(mn^2 )^3
(m^1 /^2 n^1 /^4 )^4
The brackets indicate that each letter in the bracket must
be raised to the power outside. Using law (3) of indices
gives
(mn^2 )^3
(m^1 /^2 n^1 /^4 )^4
m^1 ×^3 n^2 ×^3
m(^1 /^2 )×^4 n(^1 /^4 )×^4
m^3 n^6
m^2 n^1
Using law (2) of indices gives
m^3 n^6
m^2 n^1
=m^3 −^2 n^6 −^1 =mn^5
Problem 23. Simplify(a^3 b)(a−^4 b−^2 ), expressing
the answer with positive indices only
Using law (1) of indices givesa^3 +−^4 b^1 +−^2 =a−^1 b−^1
Using law (5) of indices givesa−^1 b−^1 =
1
a+^1 b+^1
1
ab
Problem 24. Simplify
d^2 e^2 f^1 /^2
(d^3 /^2 ef^5 /^2 )^2
expressing
the answer with positive indices only