Powers, roots and laws of indices 51
Problem 17. Find the value of23 × 35 ×( 72 )^2
74 × 24 × 3323 × 35 ×( 72 )^2
74 × 24 × 33= 23 −^4 × 35 −^3 × 72 ×^2 −^4= 2 −^1 × 32 × 70=1
2× 32 × 1 =9
2= 41
2Problem 18. Evaluate41.^5 × 81 /^3
22 × 32 −^2 /^541.^5 = 43 /^2 =√
43 = 23 = 8 , 81 /^3 =√ 3
8 = 2 ,22 = 4 , 32 −^2 /^5 =1
322 /^5=1
√ 5
322=1
22=1
4Hence,
41.^5 × 81 /^3
22 × 32 −^2 /^5=8 × 24 ×1
4=16
1= 16Alternatively,
41.^5 × 81 /^3
22 × 32 −^2 /^5=[( 2 )^2 ]^3 /^2 ×( 23 )^1 /^3
22 ×( 25 )−^2 /^5=23 × 21
22 × 2 −^2= 23 +^1 −^2 −(−^2 )= 24 = 16Problem 19. Evaluate32 × 55 + 33 × 53
34 × 54Dividing each termby theHCF (highest common factor)
of the three terms, i.e. 3^2 × 53 ,gives
32 × 55 + 33 × 53
34 × 54=32 × 55
32 × 53+33 × 53
32 × 53
34 × 54
32 × 53=3 (^2 −^2 )× 5 (^5 −^3 )+ 3 (^3 −^2 )× 50
3 (^4 −^2 )× 5 (^4 −^3 )=30 × 52 + 31 × 50
32 × 51=1 × 25 + 3 × 1
9 × 5
=28
45Problem 20. Find the value of32 × 55
34 × 54 + 33 × 53To simplify the arithmetic, each term is divided by the
HCF of all the terms, i.e. 3^2 × 53. Thus,32 × 55
34 × 54 + 33 × 53=32 × 55
32 × 53
34 × 54
32 × 53+33 × 53
32 × 53=3 (^2 −^2 )× 5 (^5 −^3 )
3 (^4 −^2 )× 5 (^4 −^3 )+ 3 (^3 −^2 )× 5 (^3 −^3 )=30 × 52
32 × 51 + 31 × 50=1 × 52
32 × 5 + 3 × 1=25
45 + 3=25
48Problem 21. Simplify7 −^3 × 34
3 −^2 × 75 × 5 −^2
expressing the answer in index form with positive
indicesSince 7−^3 =1
73,1
3 −^2= 32 and1
5 −^2= 52 ,then7 −^3 × 34
3 −^2 × 75 × 5 −^2=34 × 32 × 52
73 × 75=3 (^4 +^2 )× 52
7 (^3 +^5 )=36 × 52
78Problem 22. Simplify162 × 9 −^2
4 × 33 − 2 −^3 × 82
expressing the answer in index form with positive
indicesExpressing the numbers in terms of their lowest prime
numbers gives162 × 9 −^2
4 × 33 − 2 −^3 × 82=( 24 )^2 ×( 32 )−^2
22 × 33 − 2 −^3 ×( 23 )^2=28 × 3 −^4
22 × 33 − 2 −^3 × 26=28 × 3 −^4
22 × 33 − 23