Chapter 6: Exponents Exercise – 6.1

Question: 1

Find the values of each of the following:

(i) 132

 (ii) 73

 (iii) 34

Solution:

(i) 13² =  13 × 13

= 169

(ii) 7³ = 7 × 7 × 7

= 343

(iii) 3⁴ = 3 × 3 × 3 × 3

= 81

Question: 2

Find the value of each of the following:

(i) (-7)²

(ii) (-3)⁴

(iii)  (-5)⁵

Solution:

We know that if ‘a’ is a natural number, then

We have,

(i)  (-7)² = (-7) × (-7)

= 49

(ii) (-3)⁴ = (-3) × (-3) × (-3) × (-3)

= 81

(iii) (-5)⁵ = (-5) × (-5) × (-5) × (-5) × (-5)

= -3125

Question: 3

Simply:

(i) 3 × 10²

(ii) 2² × 5³

(iii) 3³ × 5²

Solution:

(i) 3 × 10² = 3 × 10 × 10

= 3 × 100

= 300

(ii) 2² × 5³ = 2 × 2 × 5 × 5 × 5

= 4 × 125

= 500

(iii) 3³ × 5² = 3 × 3 × 3 × 5 × 5

= 27 × 25

= 675

Question: 4

Simply:

(i)  3² × 10⁴

(ii)  2⁴ × 3²

(iii) 5² × 3⁴

Solution:

(i)  3² × 10⁴ = 3 × 3 × 10 × 10 × 10 × 10

= 9 × 10000

= 90000

(ii) 2⁴ × 3² = 2 × 2 × 2 × 2 × 3 × 3

= 16 × 9

= 144

(iii) 5² × 3⁴ = 5 × 5 × 3 × 3 × 3 × 3

= 25 × 81

= 2025

Question: 5

Simply:

(i) (-2) × (-3)³

(ii) (-3)² × (-5)³

(iii) (-2)⁵ × (-10)²

Solution:

(i) (-2) × (-3)³ = (-2) × (-3) × (-3) × (-3)

= (-2) × (-27)

= 54

(ii) (-3)² × (-5)³ = (-3) × (-3) × (-5) × (-5) × (-5)

= 9 × (-125)

= -1125

(iii) (-2)⁵ × (-10)² = (-2) × (-2) × (-2) × (-2) × (-2) × (-10) × (-10)

= (-32) × 100

= -3200

Question: 6

Simply:

(i) (3/4)²

(ii) (-2/3)⁴

(iii) (- 4/5)⁵

Solution:

Question: 7

Identify the greater number in each of the following

(i) 2⁵ or 5²

(ii) 3⁴ or 4³

(iii) 3⁵ or 5³

Solution:

(i) 2⁵ or 5²

2⁵ = 2 × 2 × 2 × 2 × 2

= 32

5² = 5 × 5

= 25

Therefore, 2⁵ 5²

(ii) 3⁴ or 4³

= 3⁴ = 3 × 3 × 3 × 3

= 81

= 4³ = 4 × 4 × 4

= 64

Therefore, 3⁴ 4³

(iii) 3⁵ or 5³

= 3⁵ = 3 × 3 × 3 × 3 × 3

= 243

= 5³ = 5 × 5 × 5

= 125

Therefore, 3⁵ 5³

Question: 8

Express each of the following in exponential form

(i) (-5) × (-5) × (-5)

Solution:

(i) (-5) × (-5) × (-5) = (-5)³

Question: 9

Express each of the following in exponential form

(i) x × x × x × x × a × a × b × b × b

(ii) (-2) × (-2) × (-2) × (-2) × a × a × a

(iii) (-2/3) × (-2/3) × x × x × x

Solution:

(i) x × x × x × x × a × a × b × b × b = x⁴a²b³

(ii) (-2) × (-2) × (-2) × (-2) × a × a × a = (-2)⁴a³

(iii) (-2/3) × (-2/3) × x × x × x = (-2/3)² x³

Question: 10

Express each of the following numbers in exponential form

(i) 512

(ii) 625

(iii) 729

Solution:

(i) 512 = 2⁹

(iii) 625 = 5⁴

(iii) 729 = 3⁶

Question: 11

Express each of the following numbers as a product of powers of their prime factors

(i) 36

(ii) 675

(iii) 392

Solution:

(i) 36 = 2 × 2 × 3 × 3

= 2² × 3²

(ii) 675 = 3 × 3 × 3 × 5 × 5

= 3³ × 5²

(iii) 392 = 2 × 2 × 2 × 7 × 7

= 2³ × 7²

Question: 12

Express each of the following numbers as a product of powers of their prime factors

(i) 450

(ii) 2800

(iii) 24000

Solution:

(i) 450 = 2 × 3 × 3 × 5 × 5

= 2 × 3² × 5²

(ii) 2800 = 2 × 2 × 2 × 2 × 5 × 5 ×7

= 2⁴ × 5² × 7

(iii) 24000 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5 × 5 × 5

= 2⁵ × 3 × 5³

Question: 13

Express each of the following as a rational number of the form p/q

(i) (3/7)²

(ii) (7/9)³

(iii) (-2/3)⁴

Solution:

Question: 14

Express each of the following rational numbers in power notation

(i) 49/64

(ii) - 64/125

(iii) -12/16

Solution:

(i) 49/64 = (7/8)²

Because 7² = 49 and 8² = 64

(ii) - 64/125 = (- 4/5)³

Because 4³ = 64 and 5³ = 125

(iii) – (1/216) = - (1/6)³

Because 1³ = 1 and 6³ = 216

Question: 15

Find the value of the following

(i) (-1/2)² × 2³ × (3/4)²

(ii) (-3/5)⁴ × (4/9)⁴ × (-15/18)²

Solution:

(i) (-1/2)² × 2³ × (3/4)² = 1/4 × 8 × 9/16

= 9/8

(ii) (-3/5)⁴ × (4/9)⁴ × (-15/18)² = 81/625 × 256/6561 × 225/324 = 64/18225

Question: 16

If a = 2 and b= 3, the find the values of each of the followimg

(i) (a + b)^a

(ii) (ab)^b

(iii) (b/a)^b

(iv) (a/b + b/a)^a

Solution:

(i) (a + b)^a = (2 + 3)²

= (5)²

= 25

(ii) (ab)^b = (2 × 3)³

= (6)³

= 216

(iii) (b/a)^b = (3/2)³

= 27/8

(iv) (a/b + b/a)^a = (2/3 + 3/2)²

= 169/36