## ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 4 Exponents and Powers Ex 4.1

Question 1.

Fill in the blanks:

(i) In the expression 3^{7}, base = …….. and exponent = ……..

(ii) In the expression (-7)^{5}, base = ………. and exponent = …….

(iii) In the expression \(\left( \frac { 2 }{ 5 } \right) ^{ 11 }\), base = …… and exponent = …….

(iv) If base is 6 and exponent is 8, then exponential form = ……..

Solution:

(i) In the expression 3^{7}, base = 3 and exponent = 7.

(ii) In the expression (-7)^{5}, base = -7 and exponent = 5.

(iii) In the expression \(\left( \frac { 2 }{ 5 } \right) ^{ 11 }\), base = -7 and exponent = 11.

(iv) If base is 6 and exponent is 8, then exponential form = 6^{8}.

Question 2.

Find the value of the following:

(i) 2^{6}

(ii) 5^{5}

(v) (-6)^{4}

(vi) \(\left( \frac { 2 }{ 3 } \right) ^{ 4 }\)

(v) \(\left( \frac { -2 }{ 3 } \right) ^{ 5 }\)

(vi) (-2)^{9}

Solution:

(i) 2^{6} = 2 × 2 × 2 × 2 × 2 × 2 = 64

(ii) 5^{5} = 5 × 5 × 5 × 5 × 5 = 3125

(iii) (-6)^{4} = (-6) × (-6) × (-6) × (-6) = 1296

(vi) (-2)^{9} = (-2) × (-2) × (-2) × (-2) × (-2) × (-2) × (-2) × (-2) × (-2) = -512

Question 3.

Express the following in the exponential form:

(i) 6 × 6 × 6 × 6 × 6

(ii) t × t × t

(iii) 2 × 2 × a × a × a × a

(iv) a × a × a × c × c × c × c × d

Solution:

(i) 6 × 6 × 6 × 6 × 6 = 6^{5}

(ii) t × t × t = t^{3}

(iii) 2 × 2 × a × a × a × a = 2^{2} × a^{4} = 2^{2}a^{4}

(iv) a × a × a × c × c × c × c × d = a^{3} × c^{4} × d = a^{3}c^{4}d

Question 4.

Simplify the following:

(i) 7 × 10^{3}

(ii) ^{25} × 9

(iii) 3^{3} × 10^{4}

Solution:

(i) 7 x 10^{3} = 7 × 10 × 10 × 10 = 7 × 1000 = 7000

(ii) 2^{5} × 9 = 2 × 2 × 2 × 2 × 2 × 9 = 32 × 9 = 288

(iii) 3^{3} × 10^{4} = 3 × 3 × 3 × 10 × 10 × 10 × 10 = 27 × 10000 = 270000

Question 5.

Simplify the following:

(i) (-3) × (-2)^{3}

(ii) (-3)^{2} × (-5)^{2}

(iii) (-2)^{3} × (-10)^{4}

(iv) (-1)^{9}

(v) 25^{2} × (-1)^{31}

(vi) 4^{2} × 3^{3} × (-1)^{122}

Solution:

(i) (-3) × (-2)^{3} = (-3) × (-2) × (-2) × (-2) = 24

(ii) (-3)^{2} × (-5)^{2} = (-3) × (-3) × (-5) × (-5) = 9 × 25 = 225

(iii) (-2)^{3} × (-10)^{4} = (-2) × (-2) × (-2) × 10 × 10 × 10 × 10 = -8 × 10000 = -80000

(iv) (-1)^{9} = -1 (∵ 9 is odd)

(v) 25^{2} × (-1)^{31} = 25 × 25 × (-1) (∵ 31 is odd)

= 625 × (-1) = -625

(vi) 4^{2} × 3^{3} × (-1)^{122}

= 4 × 4 × 3 × 3 × (1) = 144 × 1 = 144 (∵ 122 is even)

Question 6.

Identify the greater number in each of the following:

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

(ii) 7^{3} or 3^{7}

(iii) 4^{5} or 5^{4}

(iv) 2^{10} or 10^{2}

Solution:

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

4^{3} = 4 × 4 × 4 = 64

3^{4} = 3 × 3 × 3 × 3 = 81

3^{4} is greater

(ii) 7^{3} or 3^{3}

7^{3} = 7 × 7 × 7 = 343

3^{7} = 3 × 3 × 3 × 3 × 3 × 3 × 3 = 2187

3^{7} is greater

(iii) 4^{5} or 5^{4}

4^{5} = 4 × 4 × 4 × 4 × 4 = 1024

5^{4} = 5 × 5 × 5 × 5 = 625

45 is greater

(iv) 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} is greater

Question 7.

Write the following numbers as powers of 2:

(i) 8

(ii) 128

(iii) 1024

Solution:

(i) 8 = 2 × 2 × 2 = 2^{3}

(ii) 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^{7}

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

Question 8.

To what power (-2) should be raised to get 16?

Solution:

16 = (-2) × (-2) × (-2) × (-2) = (-2)^{4}

Power = 4

Question 9.

Write the following numbers as powers of (-3):

(i) 9

(ii) -27

(iii) 81

Solution:

(i) 9 = (-3) × (-3) = (-3)^{2}

(ii) -27 = (-3) × (-3) × (-3) = (-3)^{3}

(iii) 81 = (-3) × (-3) × (-3) × (-3) = (-3)^{4}

Question 10.

Find the value of x in each of the following:

(i) 7^{x} = 343

(ii) 3^{x} = 729

(in) (-8)^{x} = -512

(iv) (-4)^{x} = -1024

(v) \(\left( \frac { 2 }{ 5 } \right) ^{ x }\) = \(\frac { 32 }{ 3125 }\)

(vi) \(\left( \frac { -3 }{ 4 } \right) ^{ x }\) = \(\frac { -243 }{ 1024 }\)

Solution:

Question 11.

Write the prime factorization of the following numbers in the exponential form:

(i) 72

(ii) 360

(iii) 405

(iv) 540

(v) 2280

(vi) 3600

(vii) 4725

(viii) 8400

Solution:

(i) 72 = 2 × 2 × 2 × 3 × 3 = 2^{3} × 3^{2}

(ii) 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2^{3} × 3^{2} × 5

(iii) 405 = 3 × 3 × 3 × 3 × 5 = 3^{4} × 5^{1}

(iv) 540 = 2 × 2 × 3 × 3 × 3 × 5 = 2^{2} × 3^{3} × 5^{1}

(v) 2280 = 2 × 2 × 2 × 3 × 5 × 19 = 2^{3} × 3^{1} × 5^{1 }× 19^{1}

(vi) 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 = 2^{4} × 3^{2 }× 5^{2}

(vii) 4725 = 3 × 3 × 3 × 5 × 5 × 7 = 3^{3} × 5^{2} × 7

(viii) 8400 = 2 × 2 × 2 × 2 × 3 × 5 × 5 × 7 = 2^{4} × 3^{1} × 5^{2} × 7^{1}