## MP Board Class 7th Maths Solutions 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 × 2 × 2 × 2 × 2 × 2 = 64

(ii) 9^{3} = 9 × 9 × 9 = 729

(iii) 11^{2} = 11 × 11= 121

(iv) 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}

(ii) 343 = 7 × 7 × 7 = 7^{3}

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

(iv) 3125 = 5 × 5 × 5 × 5 × 5 = 5^{5}

Question 4.

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

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

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

(iii) 2^{5} or 8^{2}

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

(v) 2^{10} or 100^{2}

Solution:

(i) 4^{3} = 4 × 4 × 4 = 64 and 3^{4} = 3 × 3 × 3 × 3 = 81

As 81 > 64, therefore 3^{4} > 4^{3}

(ii) 5^{3} = 5 × 5 × 5 = 125,

3^{5} = 3 × 3 × 3 × 3 × 3 = 243

As 243 > 125, therefore, 3^{5} > 5^{3}

(iii) 2^{8} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256,

8^{2} = 8 × 8 = 64

As 256 > 64, therefore, 2^{8} > 8^{2}

(iv) 2^{100} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

⇒ 2^{100} = (2^{10})^{10} = 1024 × 1024 × 1024 × 1024 × 1024 × 1024 ×1024 ×1024 × 1024 × 1024 = (1024)^{10}

and 100^{2} = 100 × 100 = 10000

As (1024)^{10} > 10000, therefore 2^{100} > 100^{2}

(v) 2^{10} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

and 10^{2} = 10 × 10 = 100

As 1024 > 100, therefore 2^{10} > 10^{2}

Question 5.

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

(i) 648

(ii) 405

(iii) 540

(iv) 3600

Solution:

(i) 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2^{3} × 3^{4}

(ii) 405 = 3 × 3 × 3 × 3 × 5 = 3^{4} ×5

(iii) 540 = 2 × 2 × 3 × 3 × 3 × 5 = 2^{2} × 3^{3} × 5

(iv) 3600 = 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) 32 × 104

Solution:

(i) 2 × 10^{3} = 2 × 10 × 10 × 10 = 2 × 1000 = 2000

(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 = 3 × 256 = 768

(v) 0 × 10^{2} = 0 × 10 × 10 = 0

(vi) 5^{2} × 3^{3} = 5 × 5 × 3 × 3 × 3 = 25 × 27 = 675

(vii) 2^{4} × 3^{4} = 2 × 2 × 2 × 2 × 3 × 3 = 16 × 9 = 144

(viii) 3^{2} × 10^{4} = 3 × 3 × 10 × 10 × 10 × 10 = 9 × 10000 = 90000

Question 7.

Simplify:

(i) (-4)^{3}

(ii) (-3) × (-2)^{3}

(iii) (-3)^{2} × (-5)^{2}

(iv) (-2)^{3} × (-10)^{3}

Solution:

(i) (-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) = (-8) × (-1000) = 8000

Question 8.

Compare the following numbers:

(i) 2.7 × 10^{12}; 1.5 × 10^{8}

(ii) 4 × 10^{14}; 3 × 10^{17}

Solution:

(i) 2.7 × 10^{12}; 1.5 × 10^{8}

Since, 10^{12} > 10^{8}

∴ 2.7 × 10^{12} > 1.5 × 10^{8}

(ii) 4 × 10^{14}; 3 × 10^{17}

Since, 10^{14} < 10^{17}

∴ 4 × 10^{14} < 3 × 10^{17}