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Example 3: Express the following in exponential form or power notation:

(a) 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 (b) (−2) × (–2) × (−2) × (–2) × (−2)
(c) 4 × 4 × 4 × 3 × 3 × 3 × 3 (d) (−p) × (–p) × (−p) × (–q) ×(−q) × (–r) × (− r) × (–r)

Solution: (a) 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 = 5 8
(b) (−2) × (–2) × (−2) × (–2) × (−2) = (–2) 5

3
(c) 4 × 4 × 4 × 3 × 3 × 3 × 3 = 4 × 3 4
3
(d) (−p) × (–p) × (−p) × (–q) × (−q) × (–r) × (−r) × (–r) = (–p) × (–q) × (–r) = (–p) q (–r)
2
3 2
3
3
Example 4: Evaluate of the following:
(a) 6 4 (b) (−3) 8 (c) (–10) 5 (d) (–1) 289
4
Solution: (a) 6 = 6 × 6 × 6 × 6 = 1296
8
n
n
8
(b) (−3) = 3 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 6561 [Q (−x) = x , if n is an even number]
n
5
5
(c) (–10) = –(10) = –100000 [Q (−x) = –x , if n is an odd number]
n
n
(d) (–1) 289 = –(1) 289 = –1 [Q (−x) = –x , if n is an odd number]
n
Example 5: Compare the following:
2
2
(a) 3 , 2 3 (b) (–5) , (–2) 5
2
Solution: (a) 3 = 3 × 3 = 9
2 = 2 × 2 × 2 = 8
3
Think and Answer
Clearly, 9 > 8
Which one is greater:
2
So, 3 > 2 3 2 or 8 ?
8
2
(b) (–5) = (–5) × (–5) = 25
2
(–2) = (–2) × (–2) × (–2) × (–2) × (–2) = –32
5
Clearly, 25 > –32
2
So, (–5) > (–2) 5
Example 6: Evaluate the following:
2
(a) 5 × 2 4 (b) (−3) × (−2) 3 (c) (–1) (–4) 5 3
4
2
4
Solution: (a) 5 × 2 = 5 × (2 × 2 × 2 × 2) = 5 × 16 = 80
n
n
2
2
(b) (−3) × (−2) = 3 × (−2) 3 [Q (−3) = 3 , as (−x) = x , if n is an even number]
2
3
2
n
n
3
3
2
= −3 × 2 3 [Q (−2) = –2 , as (−x) = −x , if n is an odd number]
= − (3 × 3) × (2 × 2 × 2) = −9 × 8 = −72
3
4
3
2
(c) (–1) (–4) 5 = 1 × 4 × 5 = 1 × 4 × 4 × 4 × 4 × 5 × 5× 5
4
n
4
4
2
[Q(−1) = 1 and (−4) = 4 , as (−x) = x , if n is an even number]
n
= 1 × 256 × 125 = 32,000
Example 7: Express: Think and Answer
(a) 128 as power of 2 (b) 729 as power of 3 If a = a × a × a × a × … n times,
n
2 3
(c) 343 as power of 7 then what is the value of (5 ) ?
81 Exponents and Powers