50 Basic Engineering Mathematics
(b) ( 3 × 35 )÷( 32 × 33 )=3 × 35
32 × 33=3 (^1 +^5 )
3 (^2 +^3 )
=36
35= 36 −^5 = 31 = 3Problem 12. Simplify (a)( 23 )^4 (b)( 32 )^5 ,
expressing the answers in index formFrom law (3):
(a) ( 23 )^4 = 23 ×^4 = 212
(b) ( 32 )^5 = 32 ×^5 = 310Problem 13. Evaluate:( 102 )^3
104 × 102From laws (1) to (4):( 102 )^3
104 × 102=10 (^2 ×^3 )
10 (^4 +^2 )=106
106= 106 −^6 = 100 = 1Problem 14. Find the value of (a)23 × 24
27 × 25
(b)( 32 )^3
3 × 39From the laws of indices:(a)23 × 24
27 × 25=2 (^3 +^4 )
2 (^7 +^5 )=27
212= 27 −^12= 2 −^5 =1
25=1
32(b)( 32 )^3
3 × 39=32 ×^3
31 +^9=36
310= 36 −^10= 3 −^4 =1
34=1
81Problem 15. Evaluate (a) 4^1 /^2 (b) 16^3 /^4 (c) 27^2 /^3
(d) 9−^1 /^2(a) 4^1 /^2 =√
4 =± 2
(b) 16^3 /^4 =√ 4
163 =( 2 )^3 = 8
(Note that it does not matter whether the 4th root
of 16 is found first or whether 16 cubed is found
first – the same answer will result.)
(c) 27^2 /^3 =√ 3
272 =( 3 )^2 = 9(d) 9−^1 /^2 =1
91 /^2=1
√
9=1
± 3=±1
3Now try the following Practice ExercisePracticeExercise 30 Lawsof indices
(answers on page 342)
Evaluate the following without the aid of a
calculator.- 2^2 × 2 × 24 2. 3^5 × 33 × 3
in index form
3.27
234.33
35- 7^0 6.
23 × 2 × 26
277.10 × 106
105- 10^4 ÷ 10
9.103 × 104
109- 5^6 × 52 ÷ 57
- (7^2 )^3 in index form 12. (3^3 )^2
13.37 × 34
35in 14.( 9 × 32 )^3
( 3 × 27 )^2in
index form index form15.( 16 × 4 )^2
( 2 × 8 )^316.5 −^2
5 −^417.
32 × 3 −^4
3318.
72 × 7 −^3
7 × 7 −^419.23 × 2 −^4 × 25
2 × 2 −^2 × 2620.5 −^7 × 52
5 −^8 × 53Here are some further worked examples using the laws
of indices.Problem 16. Evaluate33 × 57
53 × 34The laws of indices only apply to termshaving the
same base. Grouping terms having the same base and
then applying the laws of indices to each of the groups
independently gives
33 × 57
53 × 34=33
34×57
53= 3 (^3 −^4 )× 5 (^7 −^3 )= 3 −^1 × 54 =54
31=625
3= 2081
3