Chapter 13 Exponents and Powers Exercise 13.1

Chapter 13 Exponents and Powers Exercise 13.1

Question 1.
Find the value of
(i) 2⁶
(ii) 9³
(iii) 11²
(iv) 5⁴
Solution:
(i) 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64
(ii) 9³ = 9 × 9 × 9 = 729
(iii) 11² = 11 × 11 = 121
(iv) 5⁴ = 5 × 5 × 5 × 5 = 625

Question 2.
Exress the following in exponential form:
(i) 6 × 6 × 6 × 6
(ii) t × t
(iii) b × b × b × b
(iv) 5 × 5 × 7 × 7 × 7
(v) 2 × 2 × a × a
(vi) a × a × a × c × c × c× c × d
Solution:
(i) 6 × 6 × 6 × 6 = 6³
(ii) t × t = t²
(iii) b × b × b × b = b⁴
(iv) 5 × 5× 7 × 7 × 7 = 5² × 7³ = 5² · 7³
(v) 2 × 2 × a × a = 2² × a² = 2² · a²
(vi) a × a ×a × c × c × c × c × d = a³ × c⁴ × d = a³ · c⁴ · d

Question 3.
Express each of the following numbers using exponential notation:
(i) 512
(ii) 343
(iii) 729
(iv) 3125
Solution:

Question 4.
Identify the greater number, wherever possible, in each of the following?
(i) 4³ or 3⁴
(ii) 5³ or 3⁵
(iii) 2⁸ or 8²
(iv) 100² or 2¹⁰⁰
(v) 2¹⁰ or 10²
Solution:
(i) 4³ or 3⁴
4³ = 4 × 4 × 4 = 64,
3⁴ = 3 × 3 × 3 × 3 = 81
Since 81 > 64
∴ 34 is greater than 43.

(ii) 5³ or 3⁵
5³ = 5 × 5 × 5 = 125
3⁵ = 3 × 3 × 3 × 3 × 3 = 243
Since 243 > 125
∴ 35 is greater than 53.

(iii) 2⁸ or 8²
2⁸ =2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
8² = 8 × 8 = 64
Since 256 > 64
∴ 28 is greater than 28.

(iv) 100² or 2¹⁰⁰
100² = 100 × 100 = 10000
2¹⁰⁰ = 2 × 2 × 2 × … 100 times
Here 2 × 2 × 2 ×2 × 2 × 2 × 2 ×2 × 2 × 2 × 2 × 2 × 2 × 2 = 214 = 16384
Since 16384 > 10,000
∴ 2100 is greater than 1002.

(v) 2¹⁰ or 10²
2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024
10² = 10 × 10 = 100
Since 1024 > 100
∴ 210 is greater than 102.

Question 5.
Express each of the following as the product of powers of their prime
(i) 648
(ii) 405
(iii) 540
(iv) 3600
Solution:

Question 6.
Simplify:
(i) 2 × 10³
(ii) 7² × 2²
(iii) 2³ × 5
(iv) 3 × 4⁴
(v) 0 × 10²
(vi) 5² × 3³
(vii) 2⁴ × 3²
(viii) 3² × 10⁴
Solution:
(i) 2 × 10³ = 2 × 10 × 10 × 10 = = 2000
(ii) 7² × 2² = = 7 × 7 × 2 × 2 = 196
(iii) 2³ × 5 = 2 × 2 × 2 × 5 = 40
(iv) 3 × 4⁴ = 3 × 4 × 4 × 4 × 4 = 768
(v) 0 × 10² = 0 × 10 × 10 = = 0
(vi) 5² × 3³ = 5 × 5 × 3 × 3 × 3 = 675
(vii) 2⁴ × 3² = 2 × 2 × 2 × 2 × 3 × 3 = 144
(viii) 3² × 10⁴ = 3 × 3 × 10 × 10 × 10 × 10 = 90000

Question 7.
Simplify:
(i) (-4)³
(ii) (-3) × (-2)³
(iii) (-3)² × (-5)²
(iv) (-2)³ × (-10)³
Solution:

Question 8.
Compare the following:
(i) 2.7 × 10¹²; 1.5 × 10⁸
(ii) 4 × 10¹⁴; 3 × 10¹⁴
Solution:
(i) 2.7 × 10¹²; 1.5 × 10⁸
Here, 10¹² > 10⁸
∴ 2.7 × 10¹²> 1.5 × 10⁸
(ii) 4 × 10¹⁴; 3 × 10¹⁷
Here, 10¹⁷ > 10¹⁴
∴ 4 × 10¹⁴ < 3 × 10¹⁷