Page 85 - Start Up Mathematics_7

Example 4: Identify the greater number, wherever possible, in each of the following.

8
2
3
7
2
(a) 7 or 3 (b) 2 or 8 (c) 100 or 2 100
3
Solution: (a) 7 or 3 7
3
7 = 7 × 7 × 7 = 343
7
and 3 = 3 × 3 × 3 × 3 × 3 × 3 × 3 = 2,187
3
∴ 7 < 3 7
2
8
(b) 2 or 8
8
2 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
2
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
4
2
(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
Solution: (a) 2 × 10 = 2 × 10 × 10 × 10 × 10 = 20,000
4
(b) 0 × 7 = 0
3
4
(c) 3 × 10 = 3 × 3 × 3 × 10 × 10 × 10 × 10 = 2,70,000
2
3
4
3
Example 7: Simplify: (a) (–3) (b) (–3) × (–2) (c) (–2) × (–10) 5
4
Solution: (a) (–3) = (–3) × (–3) × (–3) × (–3) = 81
2
3
(b) (–3) × (–2) = (–3) × (–3) × (–2) × (–2) × (–2) = 9 × –8 = –72
5
3
(c) (–2) × (–10) = (–2) × (–2) × (–2) × (–10) × (–10) × (–10) × (–10) × (–10)
= (–8) × (–1,00,000) = 8,00,000
EXERCISE 5.1
4
5
5
1. Find the value of: (a) 2 (b) 3 (c) 7 (d) 11 4
2. Write the following in exponential form:
(a) 7 × 7 × 7 × 7 × 7 (b) 2 × 2 × 3 × 3 × 3 × 3 (c) x × x × x × y × y × z × z
3. Express each of the following numbers using exponential notation:
(a) 12,500 (b) 27,000 (c) –343 (d) 62,500
4. Compare the following:
3
3
4
3
5
4
6
(a) 3 and 4 (b) (–2) and 1 (c) 3 and 5 (d) 2 and 6 2
5. Evaluate the value of each of the first five natural numbers raised to the power four.
6. To what power should (–2) be raised to get 64?
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