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## Karnataka State Syllabus Class 7 Maths Chapter 13 Exponents and Powers Ex 13.1

Question 1.

Find the value of :

(i) 2^{6}

(ii) 9^{3}

(iii) 11^{2}

(iv) 5^{4}

Solution:

i) 2^{6}

2^{6 }= 2 × 2 × 2 × 2 × 2 × 2 = 64

ii) 9^{3}

9^{3 }= 9 × 9 × 9 = 729

iii) 11^{2}

11^{2 }= 11 × 11 = 121

iv) 5^{4}

5^{4 }= 5 × 5 × 5 × 5 = 625.

Question 2.

Express 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 = 6^{4}

ii) t × t = t^{2}

iii) b × b × b × b = b^{4}

iv) 5 × 5 × 7 × 7 × 7 = 5^{2} × 7^{3}

v) 2 × 2 × a × a = 2^{2} × a^{2}

vi) a × a × a × c × c × c × c × d = a^{3} × c^{4} × d

Question 3.

Express each of the following numbers using exponential notation :

(i) 512

(ii) 343

(iii) 729

(iv) 3125

Solution:

i) 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^{9}

The exponential notation of 512 is 2^{9}

ii) 343 = 7 × 7 × 7 = 7^{3}

∴ The exponential notation of 343 is 7^{3}

iii) 729 = 3 × 3 × 3 × 3 × 3 × 3 = 3^{6}

∴ The exponential notation of 729 is 3^{6}

iv) 3125 = 5 × 5 × 5 × 5 × 5 = 5^{5}

∴ The exponential notation of 3125 is 5^{5}

Question 4.

Identify the grater number, wherever possible, in each of the following ?

Solution:

i) 4^{3} or 3^{4}

4^{3} = 4 × 4 × 4 = 64

3^{4} = 3 × 3 × 3 × 3 = 81

∴ 3^{4} > 4^{3}

ii) 5^{3} or 3^{5}

5^{3} = 5 × 5 × 5 = 125

3^{5} = 3 × 3 × 3 × 3 × 3 = 243

∴ 3^{5} >5^{3}

iii) 2^{8} or 8^{2}

2^{8} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

8^{2} = 8 × 8 = 64

∴ 2^{8} > 8^{2}

iv) 100^{2} or 2^{100}

100^{2} = 100 × 100= 10,000

2^{8} = (2^{10})^{10} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

× 2 × 2

= (1024)^{10} = [(1024)^{2}]^{5} = [10,48,576]

= (10,48,476)^{5} > 10,000

= 2^{100} > 100^{2}

v) 2^{10} or 10^{2}

2^{10} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

10^{2} = 10 × 10 = 100

∴ 2^{10} > 10^{2}

Question 5.

Express each of the following as product of powers of their prime factors :

Solution:

i) 648

648 = 2 × 2 × 2 × 3 × 3 × 3 × 3

= 2^{3} × 3^{4}

ii) 405

405 = 5 × 3 × 3 × 3 × 3

= 5 × 3^{4}

iii) 540

540 = 2 × 2 × 3 × 3 × 3 × 5

= 2^{2} × 3^{3} × 5

iv) 3,600

3,600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

= 2^{4} × 3^{2} × 5^{2}

Question 6.

Simplify :

(i) 2 × 10^{3}

(ii) 7^{2} × 2^{2}

(iii) 2^{3} × 5

(iv) 3 × 4^{4}

(v) 0 × 10^{2}

(vi) 5^{2} × 3^{3}

(vii) 2^{4} × 3^{2}

(viii) 3^{2} × 10^{4}

Solution:

i) 2 × 10^{3}

= 2 × 10 × 10 × 10 = 2,000

ii) 7^{2} × 2^{2}

= 7 × 7 × 2 × 2

= 49 × 4 = 196

iii) 2^{3} × 5

= 2 × 2 × 2 × 5

= 8 × 5 = 40

iv) 3 × 4^{4}

= 3 × 4 × 4 × 4 × 4

= 12 × 64 = 768

v) 0 × 10^{2}

= 0 × 10 × 10 = 0

vi) 5^{2} × 3^{3}

= 5 × 5 × 3 × 3 × 3

= 25 × 27 = 675

vii) 24 × 32

=2 × 2 × 2 × 2 × 3 × 3

= 16 × 9 = 144

viii) 3^{2 × 104
= 3 × 3 × 10 × 10 × 10 × 10
= 90 × 1000 = 90,000}

Question 7.

Simplify :

Solution:

i) (-4)^{3}

(-4)^{3} = (-4) × (-4) × (-4) = -64

ii) (-3) × (-2)^{3}

= (-3) × (-2) × (-2) × (-2) = (-3) × (-8) = 24

iii) (-3)^{2} × (-5)^{2}

= (-3) × (-3) × (-5) × (-5)

= 9 × 25 = 225

iv) (-2)^{3} × (-10)^{3}

= (-2) × (-2) × (-2) × (-10) × (-10) × (-10)

= +4 × -2 × 100 × -10

= -8 × -1000 = 8000

Question 8.

Compare the following numbers :

Solution:

i) 2 : 7 × 10^{12} ; 1.5 × 10^{8}

= 2.7 × 10^{12}

= 2.7 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

= 2.70000000000000

= 27 × 10^{11}

1.5 × 10^{7}

= 1.5 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

= 1.5 × 10^{7}

∴ 2.7 × 10^{12} >1.5 × 10^{8}

ii) 4 × 10^{14} ; 3 × 10^{17}

=4 × 10^{14} = 4 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

= 4 × 10^{14}

3 × 10^{7} = 3 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

∴ 3 × 10^{17} > 4 × 10^{14}