Section outline

    • Solutions to Ex 11.1
      Q1. Find the value of:

      Ans:

      (i) 26 = 2 x 2 x 2 x 2 x 2 x 2 = 64

      (ii) 93 = 9 x 9 x 9 = 729

      (iii) 112 = 11 x 11 = 121

      (iv) 54 = 5 x 5 x 5 x 5 = 625

      Q2. Express the following in exponential form:

      Ans:

      (i) 6 x 6 x 6 x 6 = 64

      (ii) t x t = t2

      (iii) b x b x b x b = b4

      (iv) 5 x 5 x 7 x 7 x 7 = 52 x 73

      (v) 2 x 2 x a x a = 22 x a2

      (vi) a x a x a x c x c x c x c x d = a3 x c4 x d

      Q3. Express each of the following numbers using exponential notation:

      Ans:

      (i) 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 29

      (ii) 343 = 7 x 7 x 7 = 73

      (iii) 729 = 3 x 3 x 3 x 3 x 3 x 3 = 36

      (iv) 3125 = 5 x 5 x 5 x 5 x 5 = 55

      Q4. Identify the greater number, wherever possible, in each of the following?

      Ans:

      (i) 43 or 34

      43 = 4 x 4 x 4 = 64

      34= 3 x 3 x 3 x 3 = 81

      34 is the greater number.

      (ii) 53 or 35

      53 = 5 x 5 x 5 = 125

      35= 3 x 3 x 3 x 3 x 3 = 243

      35 is the greater number.

      (iii) 28 or 82

      28 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

      82 =8 x 8 = 64

      28 is the greater number.

      (iv) 1002 or 2100

      1002 = 100 x 100 = 10000

      2100 = (1024)10 

      2100 is the greater number.

      (v) 210 or 102

      210 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2= 1024

      102 = 10 x 10 = 100

      210 is the greater number.

      Q5. Express each of the following as product of powers of their prime factors:

      Ans:

      (i) 648 = 2 x 2 x 2 x 3 x 3 x 3 x 3 = 23 x 34

      (ii) 405 = 3 x 3 x 3 x 3 x 5 = 34 x 5

      (iii) 540 = 2 x 2 x 3 x 3 x 3 x 5 = 22 x 33 x 5

      (iv) 3600 = 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 = 24 x 32 x 52

      Q6. Simplify:

      Ans:

      (i) 2 x 103 = 2 x 10 x 10 x 10 = 2000

      (ii) 72 x 22 = 7 x 7 x 2 x 2 = 196

      (iii) 23 x 5 = 2 x 2 x 2 x 5 = 40

      (iv) 3 x 44 = 3 x 4 x 4 x 4 x 4 = 768

      (v) 0 x 102 = 0

      (vi)  52 x 33 = 5 x 5 x 3 x 3 x 3 = 675

      (vii) 24 x 32 = 2 x 2 x 2 x 3 x 3 = 144

      (viii) 32 x 104 = 3 x 3 x 10 10 x 10 x 10 = 90000

      Q7. Simplify:

      Ans:

      (i) (– 4)3 = (– 4) x (– 4) x (– 4) = – 64

      (ii) (–3) x (–2)3 = (–3) x (–2) x (–2) x (–2) = 24

      (iii) (–3)2 x (–5)2 = (–3) x (–3) x (–5) x (–5) = 225

      (iv) (–2)3 x (–10)3 = (–2) x (–2) x (–2) x (–10) x (–10) x (–10) = 8000

      Q8. Compare the following numbers:

      Ans:

      (i) 2.7 x 1012 ; 1.5 x 108

      Comparing the exponents,

      1012 > 108

      Thus, 2.7 x 1012 > 1.5 x 108

      (ii) 4 x 1014 ; 3 x 1017

      Comparing the exponents,

      1014< 1017

      Thus, 4 x 1014 < 3 x 1017

       

    • Solutions to Ex. 11.2
      Q1. Using laws of exponents, simplify and write the answer in exponential form:

      Ans:

      (i)  32 x 34 x 38 = 3(2 + 4 + 8) = 314

      (ii) 615 ÷ 610 = 6(15 - 10) = 65

      (iii) a3 x a2 = a(3 + 2) = a5

      (iv)  7x x 72 = 7 x + 2

      (v) (52)3 ÷ 53= 5(2 x 3) ÷ 53

      = 5(6 - 3) = 53

      (vi) 25 × 55 = (2 x 5)5 = 105 

      (vii) a4 × b4 = (ab)4 

      (viii) (34)3 = 3(4 x 3) = 312

      (ix) (220 ÷ 215) x 23

      = 2 (20 - 15) x 23

      = 25 x  23 = 2 (5 = 3) = 28

      (x) 8t ÷ 82 = 8t - 2

      Q2. Simplify and express each of the following in exponential form:

      Ans:

      (i) \( \frac{2^3 \times 3^4 \times 4}{3 \times 32} \) 

      = \( \frac{2^3 \times 3^4 \times 2^2}{3 \times 2^5} \) = \( \frac{2^{3+2} \times 3^4}{3 \times 2^5} \) 

      = \( \frac{2^5 \times 3^4}{3 \times 2^5} \) = 3(4 - 1) = 33

      (ii) ((52)3 x 54) ÷ 57 = (52 x 3 x 54) ÷ 57

      = (56 x 54) ÷ 57 = (56 + 4 ) ÷ 5 = 510 ÷ 5

      = 510 - 7 = 53

      (iii) 254 ÷ 53 = (52)4 ÷ 53

      = 52 × 4 ÷ 53 = 58 ÷ 53

      = 58-3 = 55

      (iv) \( \frac{3 \times 7^2 \times 11^8}{21 \times 11^3} \) 

      = \( \frac{3 \times 7^2 \times 11^8}{3 \times 7 \times 11^3} \) 

      = 72-1 x 118 – 3 = 7 x 115

      (v) \( \frac{3^7}{3^4 \times 3^3} \) = ( \frac{3^7}{3^{4 +3}} \)

      = \( \frac{3^7}{3^7} \) = 37 - 7 = 1 or 30

      (vi) 20 + 30 + 40 = 1 + 1 + 1 = 3

      (vii) 20 x 30 x 40 = 1 x 1 x 1 = 1

      (viii) (30 + 20) × 50 = (1 + 1) x 1

      = 2 x 1 = 1

      (ix) \( \frac{2^8 \times a^5}{4^3 \times a^3} \) 

      As, (4)3 = (22)3 = 22 x 3 = 26

      \( \frac{2^8 \times a^5}{4^3 \times a^3} \) = \( \frac{2^8 \times a^5}{2^6 \times a^3} \) 

      = 28 – 6 x a5 – 3 = 2a2

      (x) (\( \frac{a^5}{a^3} \)) x a8

      = (a5 -3) x a8 = a2 x a8

      = a2 + 8 = a10

      (xi) \( \frac{4^5 \times a^8 \times b^3}{4^5 \times a^5\times b^2} \)

      = 45 – 5 x a8 – 5 x b3 – 2 = 40 x a3 x b

      = a3b

      (xii) (23 × 2)2 = (23 + 1)2

      = (24)2 = 24 × 2 = 28

      Q3. Say true or false and justify your answer:

      Ans:

      (i) 10 x 1011 = 10011

      LHS = 10 x 1011= 101 + 11

      LHS = 1012

      RHS = 10011 = (102)11 = 102 x 11

      RHS = 1022

      As, 1012 ≠ 1022

      The statement 10 x 1011 = 10011 is false.

      (ii) 23 > 52

      LHS = 23 = 2 x 2 x 2

      LHS = 8

      RHS = 52 = 5 x 5

      RHS = 25

      As, 8 ≠ 25

      The statement 23 > 52 is false.

      (iii) 23 x 32 = 65

      LHS =  23 x 32 = 2 x 2 x 2 x 3 x 3

      LHS = 72

      RHS = 65 = 6 x 6 x 6 x 6 x 6

      RHS = 7776

      As, 72 ≠ 7776

      The statement 23 x 32 = 65 is false.

      (iv) 30 = (1000)0

      LHS = 30 = 1

      RHS = (1000)0 = 1

      LHS = RHS

      Thus, 30 = (1000)0 is true.

      Q4. Express each of the following as a product of prime factors only in exponential form:

      Ans:

      (i) 108 × 192

      Prime factors of 108 = 2 x 2 x 3 x 3 x 3 = 22 x 33

      Prime factors of 192 =  2 x 2 x 2 x 2 x 2 x 2 x 3 = 26 x 3

      Thus, 108 × 192 = (22 x 33) x (26 x 3)

      = 22 + 6 x 33 + 1 = 28 x 34

      (ii) 270 = 2 x 3 x 3 x 3 x 5 = 2 x 33 x 5

      (iii) 729 × 64

      Prime factors of 729 = 3 x 3 x 3 x 3 x 3 x 3 = 36

      Prime factors of 64 =  2 x 2 x 2 x 2 x 2 x 2 = 26 

      Thus, 729 × 64 = 36 x 26 

      (iv) 768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 = 28 x 3

      Q5. Simplify:

      Ans:

      (i) \( \frac{({2^5})^2 \times 7^3}{8^3 \times 7} \) 

      = \( \frac{2^{(5 \times2)} \times 7^3}{({2^3})^3 \times 7} \) 

      = \( \frac{2^{10} \times 7^3}{2^9 \times 7} \) = 2(10-9) x 7(3-1)

      = 2 x 72 = 2 x 7 x 7 = 98

      (ii) \( \frac{25 \times 5^2 \times t^8}{10^3 \times t^4} \) 

      = \( \frac{5^2 \times 5^2 \times t^8}{{(5 \times 2)}^3 \times t^4} \) 

      = \( \frac{5^{2 + 2} \times t^8}{5^3 \times 2^3 \times t^4} \) 

      = \( \frac{5^{4 - 3} \times t^{8 - 4}}{2^3} \) = \( \frac{5t^4}{8} \)

      (iii) \( \frac{3^5 \times 10^5 \times 25}{5^7 \times 6^5} \) 

      = \( \frac{3^5 \times {(5 \times 2)}^5 \times 5^2}{5^7 \times {(2 \times 3)}^5} \) 

      = \( \frac{3^5 \times 5^5 \times 2^5 \times 5^2}{5^7 \times 2^5 \times 3^5} \) 

      = 2(5 - 5) x 3(5 - 5) x 5(5 + 2 - 7) = 20 + 30 + 50 = 1

       
    • Solutions to Ex. 11.3
      Q1. Write the following numbers in the expanded forms:

      Ans:

      (i) 279404 

      = 2 x 100000 + 7 x 10000 + 9 x 1000 + 4 x 100 + 0 x 10 + 4 x 1

      = 2 x 105 + 7 x 104 + 9 x 103 + 4 x 102+ 4 x 100

      (ii) 3006194

      = 3 x 1000000 + 0 x 100000 + 0 x 10000 + 6 x 1000 + 1 x 100 + 9 x 10 + 4 x 1

      = 3 x 106 + 6 x 103 + 1 x 102+ 9 x 101 + 4 x 100

      (iii) 2806196

      = 2 x 1000000 + 8 x 100000 + 0 x 10000 + 6 x 1000 + 1 x 100 + 9 x 10 + 6 x 1

      = 2 x 105 + 8 x 104 + 6 x 103+ 1 x 102 + 9 x 101 + 6 x 100

      (iv) 120719

      = 1 x 100000 + 2 x 10000 + 0 x 1000 + 7 x 100 + 1 x 10 + 9 x 1

      = 1 x 105 + 2 x 104 + 7 x 102+ 1 x 101 + 9 x 100

      (v) 20068

      = 2 x 10000 + 0 x 1000 + 0 x 100 + 6 x 10 + 8 x 1

      = 2 x 104 + 6 x 101 + 8 x 100

      Q2. Find the number from each of the following expanded forms:

      Ans:

      (a) (8 x 10)4 + (6 x 10)3 + (0 x 10)2 + (4 x 10)1 + (5 x 10)0

      = (8 x 10000) + (6 x 1000) + (0 x 100) + (4 x 10) + (5 x 1)

      = 80000 + 6000 + 0 + 40 + 5 = 86045

      (b) (4 x 10)5 + (5 x 10)3 + (3 x 10)2 + (2 x 10)0

      = (4 x 100000) + (0 x 10000) + (5 x 1000) + (3 x 100) + (0 x 10) + (2 x 1)

      = 400000 + 0 + 5000 + 300 + 0 + 2 = 405302

      (c) (3 x 10)4 + (7 x 10)2 + (5 x 10)0

      = (3 x 10000) + (0 x 1000) + (7 x 100) + (0 x 10) + (5 x 1)

      = 30000 + 0 + 700 + 0 + 5 = 30705

      (d) (9 x 10)5 + (2 x 10)2 + (3 x 10)1

      = (9 x 100000) + (0 x 10000) + (0 x 1000) + (2 x 100) + (3 x 10) + (0 x 1)

      = 900000 + 0 + 0 + 200 + 30 + 0 = 900230

      Q3. Express the following numbers in standard form:

      Ans:

      (i) 5,00,00,000 = 5 x 107

      (ii) 70,00,000 = 7 x 106

      (iii) 3,18,65,00,000 = 3.1865 x 109

      (iv) 3,90,878 = 3.90878 x 105

      (v) 39087.8 = 3.90878 x 104

      (vi) 3908.78 = 3.90878 x 103

      Q4. Express the number appearing in the following statements in standard form.

      Ans:

      (a) 384,000,000 m = 3.84 x 108m

      (b) 300,000,000 m/s = 3 x 108m/s

      (c) 1,27,56,000 m = 1.2756 x 107m

      (d) 1,400,000,000 m = 1.4 x 109m

      (e) 100,000,000,000 stars = 1 x 1011 stars

      (f) 12,000,000,000 years = 1.2 x 1010

      (g)  300,000,000,000,000,000,000 m = 3 x 1020m

      (h) 60,230,000,000,000,000,000,000 molecules = 6.023 x 1022

      (i) 1,353,000,000 cubic km = 1.353 x 109

      (j) 1,027,000,000 = 1.027 x 109