Page 69 - ICSE Math 7

Example 3: Express each of the following numbers using exponential notation.
–8 64
(a) 1,024 (b) 243 (c) 15,625 (d) (e)
343 729
Solution: (a) 1,024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 10

(b) 243 = 3 × 3 × 3 × 3 × 3 = 3 5
(c) 15,625 = 5 × 5 × 5 × 5 × 5 × 5 = 5 6

–8 (–2) × (–2) × (–2) –2 3
(d) = =
343 7 × 7 × 7 7
64 2 × 2 × 2 × 2 × 2 × 2 2 6
(e) = =
729 3 × 3 × 3 × 3 × 3 × 3 3
Example 4: Identify the greater number, wherever possible, in each of the following.
3
2
2
8
7
(a) 7 or 3 (b) 2 or 8 (c) 100 or 2 100
3
Solution: (a) 7 or 3 7
7
3
7 = 7 × 7 × 7 = 343 and 3 = 3 × 3 × 3 × 3 × 3 × 3 × 3 = 2,187
3
∴ 7 < 3 7
8
2
(b) 2 or 8
2
8
2 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256 and 8 = 8 × 8 = 64
8
∴ 2 > 8 2
2
(c) 100 or 2 100
2
100 = 100 × 100 = 10,000
and 2 100 is a very large number
2
∴ 100 < 2 100
Example 5: Express each of the following as a product of powers of their prime factors.
(a) 216 (b) 1,080 (c) 3,600
3
Solution: (a) 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2 × 3 3
3
3
(b) 1,080 = 2 × 2 × 2 × 3 × 3 × 3 × 5 = 2 × 3 × 5
2
4
(c) 3,600 = 2 × 2 × 5 × 5 × 6 × 6 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 = 2 × 3 × 5 2
4
4
3
Example 6: Simplify: (a) 2 × 10 (b) 0 × 7 (c) 3 × 10 4
4
(d) (–3) (e) (–3) × (–2) (f) (–2) × (–10) 5
3
3
2
4
Solution: (a) 2 × 10 = 2 × 10 × 10 × 10 × 10 = 20,000
4
(b) 0 × 7 = 0
4
3
(c) 3 × 10 = 3 × 3 × 3 × 10 × 10 × 10 × 10 = 2,70,000
4
(d) (–3) = (–3) × (–3) × (–3) × (–3) = 81
3
2
(e) (–3) × (–2) = (–3) × (–3) × (–2) × (–2) × (–2) = 9 × (–8) = –72
5
3
(f) (–2) × (–10) = (–2) × (–2) × (–2) × (–10) × (–10) × (–10) × (–10) × (–10)
= (–8) × (–1,00,000) = 8,00,000
Points to remember
• A negative rational number raised to an even power is always positive.
12
32
For example, (–1) = 1; (–1) = 1
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