NCERT Solutions for class 7 Maths Chapter 13: Exponents and PowersExercise 13.11. Find the value of: (i) 26 Answer: 64 Explanation: 26 can be represented as: 2 × 2 × 2 × 2 × 2 × 2 = 64 (ii) 93 Answer: 729 Explanation: 93 can be represented as: 9 × 9 × 9 = 81 × 9 = 729 (iii) 112 Answer: 121 Explanation: 112 can be represented as: 11 × 11 = 121 (iv) 54 Answer: 625 Explanation: 54 can be represented as: 5 × 5 × 5 × 5 = 25 × 25 = 625 2. Express the following in exponential form: (i) 6 × 6 × 6 × 6 Answer: 64 Explanation: The number 6 is repeated four times. Hence, it can be represented as: 64 (ii) t × t Answer: t2 Explanation: The variable t is repeated two times. Hence, it can be represented as: t2 (iii) b × b × b × b Answer: b4 Explanation: The variable b is repeated two times. Hence, it can be represented as: b4 (iv) 5 × 5 × 7 × 7 × 7 Answer: 52 × 73 Explanation: The above expression has two numbers 5 and 7. The number 5 is repeated two times and the number 7 is repeated three times. Hence, it can be represented as: 52 × 73 (v) 2 × 2 × a × a Answer: 22 × a2 Explanation: The above expression has one number 2 and one variable a. The number 2 is repeated two times and the variable 'a' is also repeated two times. Hence, it can be represented as: 22 × a2 (vi) a × a × a × c × c × c × c × d Answer: a3 × c4 × d Explanation: The above expression has three variables a, c, and d. The variable 'a' is repeated three times, variable 'c' four times, and the variable 'd 'one time. Hence, it can be represented as: a3 × c4 × d 3. Express each of the following numbers using exponential notations: (i) 512 Answer: 29 Explanation: 512 can be represented as: = 2 × 256 = 2 × 2 × 128 = 2 × 2 × 2 × 64 = 2 × 2 × 2 × 2 × 32 = 2 × 2 × 2 × 2 × 2 × 16 = 2 × 2 × 2 × 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 The number 2 is repeated nine times. Hence, it can be written as: = 29 (ii) 343 Answer: 73 Explanation: 343 can be represented as: = 7 × 49 = 7 × 7 × 7 The number 7 is repeated three times. Hence, it can be written as: = 73 (iii) 729 Answer: 36 Explanation: 729 can be represented as: = 3 × 243 = 3 × 3 × 81 = 3 × 3 × 3 × 27 = 3 × 3 × 3 × 3 × 9 = 3 × 3 × 3 × 3 × 3 × 3 The number 3 is repeated six times. Hence, it can be written as: = 36 (iv) 3125 Answer: 55 Explanation: 3125 can be represented as: = 5 × 625 = 5 × 5 × 125 = 5 × 5 × 5 × 25 = 5 × 5 × 5 × 5 × 5 The number 5 is repeated five times. Hence, it can be written as: = 55 4. Identify the greater number, wherever possible, in each of the following? (i) 43 or 34 Answer: 34 Explanation: 43 can be represented as: 4 × 4 × 4 = 16 × 4 = 64 34 can be represented as: 3 × 3 × 3 × 3 = 9 × 3 × 3 = 27 × 3 = 81 81 > 64 Hence, 34 is greater. (ii) 53 or 35 Answer: 35 Explanation: 53 can be represented as: 5 × 5 × 5 = 25 × 5 = 125 35 can be represented as: 3 × 3 × 3 × 3 × 3 = 9 × 3 × 3 × 3 = 27 × 3 × 3 = 81× 3 = 243 243 > 125 Hence, 35 is greater. (iii) 28 or 82 Answer: 28 Explanation: 28 can be represented as: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 16 × 16 = 256 82 can be represented as: 8 × 8 = 64 256 > 64 Hence, 28 is greater. (iv) 1002 or 2100 Answer: 2100 Explanation: 2100 is greater than 1002. (v) 210 or 102 Answer: 210 Explanation: 210 can be represented as: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 16 × 16 × 4 = 1024 102 can be represented as: 10 × 10 = 100 1024 > 100 Hence, 210 is greater. 5. Express each of the following as product of powers of their prime factors: (i) 648 Answer: 23 × 34 Explanation: 648 can be represented as: = 2 × 324 = 2 × 2 × 162 = 2 × 2 × 2 × 81 = 2 × 2 × 2 × 3 × 27 = 2 × 2 × 2 × 3 × 3 × 9 = 2 × 2 × 2 × 3 × 3 × 3 × 3 The number 2 is repeated three times and the number 3 is repeated four times. Hence, it can be written as: 23 × 34 (ii) 405 Answer: 5× 34 Explanation: 405 can be represented as: = 5× 81 = 5 × 3 × 27 = 5 × 3 × 3 × 9 = 5 × 3 × 3 × 3 × 3 The number 5 is repeated one time and the number 3 is repeated four times. Hence, it can be written as: 51 × 34 = 5× 34 (iii) 540 Answer: 22 × 33 × 5 Explanation: 540 can be represented as: = 2 × 270 = 2 × 27 × 10 = 2 × 3 × 3 × 3 × 2 × 5 = 22 × 33 × 5 (iv) 3,600 Answer: 24× 32 × 52 Explanation: 3600 can be represented as: = 60 × 60 = 3 × 2 × 2 × 5 × 3 × 2 × 2 × 5 The number 3 is repeated two times, the number 2 four times, and the number 5 two times. Hence, the expression can be written as: 24× 32 × 52 6. Simplify: (i) 2 × 103 Answer: 2000 Explanation: 2 × 103 The number 10 is repeated three times. = 2 × 10 × 10 × 10 = 2 × 1000 = 2000 (ii) 72 × 22 Answer: 196 Explanation: 72 × 22 = 7 × 7 × 2 × 2 = 49 × 4 = 196 (iii) 23 × 5 Answer: 40 Explanation: 23 × 5 = 2 × 2 × 2 × 5 = 8 × 5 = 40 (iv) 3 × 44 Answer: 768 Explanation: 3 × 44 = 3 × 4 × 4 × 4 × 4 = 3 × 16 × 16 = 3 × 256 = 768 (v) 0 × 102 Answer: 0 Explanation: Any number multiplied by 0, gives 0 as the result. (vi) 52 × 33 Answer: 675 Explanation: 52 × 33 = 5 × 5 × 3 × 3 × 3 = 25 × 27 = 675 (vii) 24 × 32 Answer: 144 Explanation: 24 × 32 = 2 × 2 × 2 × 2 × 3 × 3 = 16 × 9 = 144 (viii) 32 × 104 Answer: 90000 Explanation: 32 × 104 = 3 × 3 × 10 × 10 × 10 × 10 = 9 × 10000 = 90000 7. Simplify: (i) (- 4)3 Answer: - 64 Explanation: (- 4)3 means that the integer (- 4) is repeated three times. = (- 4) × (- 4) × (- 4) = 16 × (- 4) = - 64 (ii) (-3) × (-2)3 Answer: 24 Explanation: (- 2)3 means that the integer (- 2) is repeated three times. (-3) × (-2)3 =(-3) × (-2) × (-2) × (-2) = 6 × (-2) × (-2) = -12 × (-2) = 24 (iii) (-3)2 × (-5)2 Answer: 225 Explanation: Both the integers (- 3) and (- 5) are repeated two times. (-3)2 × (-5)2 = (-3) × (-3) × (-5) × (-5) = 9 × (-5) × (-5) = (-45) × (-5) = 225 (iv) (-2)3 × (-10)3 Answer: 8000 Explanation: Both the integers (- 2) and (- 10) are repeated three times. (-2)3 × (-10)3 = (-2) × (-2) × (-2) × (-10) × (-10) × (-10) = (-8) × (-1000) = 8000 8. Compare the following numbers: (i) 2.7 × 1012 ; 1.5 × 108 Answer: 2.7 × 1012 > 1.5 × 108 Explanation: 2.7 × 1012 = 2,700,000,000,000 1.5 × 108 = 150,000,000 Hence, 2.7 × 1012 is greater. 2.7 × 1012 > 1.5 × 108 (ii) 4 × 1014 ; 3 × 1017 Answer: 4 × 1014 < 3 × 1017 Explanation: 1017 is greater than the 1014 Hence, 4 × 1014 is greater. 3 × 1017 > 4 × 1014 Or 4 × 1014 < 3 × 1017 Exercise 13.21. Using laws of exponents simplify and write the answer in exponential form: (i) 32 × 34 × 38 Answer: 312 Explanation: In the case of multiplication, the exponents are added if the base of numbers is the same. 32 × 34 × 38 = 3(2 + 4 + 8) 32 × 34 × 38 = 312 (ii) 615 ÷ 610 Answer: 65 Explanation: In the case of division, the exponents are subtracted if the base of numbers is the same. 615 ÷ 610 = 6(15 - 10) 615 ÷ 610 = 65 (iii) a3 × a2 Answer: a5 Explanation: In the case of multiplication, the exponents are added if the base of numbers is the same. a3 × a2= a(3 + 2) a3 × a2= a5 (iv) 7x ×72 Answer: 7(x + 2) Explanation: In the case of multiplication, the exponents are added if the base of numbers is the same. 7x ×72 = 7(x + 2) 7x ×72 = 7x + 2 (v) (52)3 ÷ 53 Answer: 53 Explanation: The exponents in the case of double brackets are multiplied. (52)3 = 52 x 3 (52)3 = 56 In the case of division, the exponents are subtracted if the base of numbers is the same. 56 ÷ 53 = 5(6 - 3) 56 ÷ 53 = 53 (vi) 25 × 55 Answer: 105 Explanation: 25 × 55 = (2 × 5)5 25 × 55 = (10)5 (vii) a4 × b4 Answer: (ab)4 Explanation: a4 × b4 = (a × b)4 a4 × b4 = (ab)4 a4 × b4 = ab4 (viii) (34)3 Answer: 312 Explanation: The exponents in the case of double brackets are multiplied. (34)3 = 34 x 3 (34)3 = 312 (ix) (220 ÷ 215) × 23 Answer: 28 Explanation: (220 ÷ 215) × 23 In the case of division, the exponents are subtracted if the base of numbers is the same. = 2(20 - 15) × 23 = 25 × 23 In the case of multiplication, the exponents are added if the base of numbers is the same. = 2(5 + 3) = 28 (x) 8t ÷ 82 Answer: 8t - 2 Explanation: 8t ÷ 82 In the case of division, the exponents are subtracted if the base of numbers is the same. = 8(t - 2) 2. Simplify and express each of the following in exponential form: (i) (23 × 34 × 4)/ (3 × 32) Answer: 33 Explanation: To simplify, let's convert the available numbers into exponents. 4 can be written as: 2 × 2 = 22 32 can be written as: 2 × 2 × 2 × 2 × 2 = 25 Substituting the value of 4 and 32 in the expression, (23 × 34 × 22)/ (3 × 25) In the case of division, the exponents are subtracted, and added in the case of multiplication if the base of numbers is the same. = (23 + 2 - 5 × 34 - 1) = (20 × 33) = 33 (20 = 1) (ii) ((52)3 × 54) ÷ 57 Answer: 53 Explanation: ((52)3 × 54)÷ 57 The exponents in the case of double brackets are multiplied. (52)3 = 52 x 3 (52)3 = 56 ((52)3 × 54)÷ 57 can be written as: (56 × 54)÷ 57 In the case of division, the exponents are subtracted, and added in the case of multiplication if the base of numbers is the same. = (56 + 4) ÷ 57 = (510) ÷ 57 = 510 - 7 = 53 (iii) 254 ÷ 53 Answer: 55 Explanation: To simplify, let's convert the available numbers into exponents. 25 can be written as: 5 × 5 = 52 Substituting the value of 25 in the expression, (52)4 ÷ 53 The exponents in the case of double brackets are multiplied. (52)4 = 52 x 4 (52)4 = 58 In the case of division, the exponents are subtracted if the base of numbers is the same. 58 ÷ 53 = 58 - 3 = 55 (iv) (3 × 72 × 118)/ (21 × 113) Answer: 7× 115 Explanation: To simplify, let's convert the available numbers into exponents. 21 can be written as: 3 × 7 Substituting the value of 21 in the expression, (3 × 72 × 118)/ (7 × 3 × 113) In the case of division, the exponents are subtracted, and added in the case of multiplication if the base of numbers is the same. (31 - 1 × 72 - 1 × 118 - 3) = (30 × 71 × 115) = (1× 71 × 115) = (7× 115) (v) 37/ (34 × 33) Answer: 30 or 1 Explanation: 37/ (34 × 33) In the case of multiplication, the exponents are added if the base of numbers is the same. = 37/ (34 + 3) = 37/ 37 In the case of division, the exponents are subtracted if the base of numbers is the same. = 37 - 7 = 30 = 1 (vi) 20 + 30 + 40 Answer: 3 Explanation: Any number with the exponent 0 is considered as 1. 20 = 1 30 = 1 40 = 1 Substituting the value in the expression, we get: 20 + 30 + 40 = 1 + 1 + 1 = 3 (vii) 20 × 30 × 40 Answer: 1 Explanation: Any number with the exponent 0 is considered as 1. 20 = 1 30 = 1 40 = 1 Substituting the value in the expression, we get: 20 × 30 × 40 = 1 × 1 × 1 = 1 (viii) (30 + 20) × 50 Answer: 2 Explanation: Any number with the exponent 0 is considered as 1. 20 = 1 30 = 1 50 = 1 Substituting the value in the expression, we get: (30 + 20) × 50 = (1 + 1) × 1 = 2 × 1 = 2 (ix) (28 × a5)/ (43 × a3) Answer: (2a)2 Explanation: To simplify, let's convert the available numbers into exponents. 4 can be written as: 2 × 2 = 22 Substituting the value of 4 in the expression, (28 × a5)/ ((22)3 × a3) (22)3 = 22 x 3 (22)3 = 26 = (28 × a5)/ (26× a3) In the case of division, the exponents are subtracted if the base of numbers is the same. = (28 - 6 × a5 - 3) = (22× a2) = (2a)2 (x) a5/a3 × a8 Answer: a10 Explanation: The above expression can be written as: = (a5 × a8) /a3 = (a5 + 8)/a3 = a13/a3 = a10 (xi) (45 × a8b3)/(45 × a5b2) Answer: a3b Explanation: To simplify, let's convert the available numbers into exponents. 4 can be written as: 2 × 2 = 22 Substituting the value of 4 in the expression, ((22)5 × a8b3)/( (22)5 × a5b2) (22)5 = 22 x 5 (22)5 = 210 (210 × a8b3)/(210 × a5b2) In the case of division, the exponents are subtracted, and added in the case of multiplication if the base of numbers is the same. = (210 - 10 × a8 - 5 b3 - 2) = (20× a3b1) = a3b (xii) (23 × 2)2 Answer: 28 Explanation: (23 × 2)2 = 23 x 2 × 22 = 26 × 22 In the case of multiplication, the exponents are added if the base of numbers is the same. = 26 + 2 = 28 3. Say true or false and justify your answer: (i) 10 × 1011 = 10011 Answer: False; 10 × 1011 = 1012; 10011 = 1022 Explanation: The expression can be written as: 10 × 1011 = 101 × 1011 In the case of multiplication, the exponents are added if the base of numbers is the same. = 101 + 11 = 1012 Thus, the correct expression is: 10 × 1011 = 1012 10011 can be written as: (102)11 = 102 x 11 = 1022 Thus, 10011 = 1022 (ii) 23 > 52 Answer: False; 23 < 52 Explanation: 23 can be represented as: 2 × 2 × 2 = 8 52 can be represented as: 5 × 5 = 25 8 < 25 Thus, 23 < 52 (iii) 23 × 32 = 65 Answer: False; 65 = 25 × 35 Explanation: 23 can be represented as: 2 × 2 × 2 = 8 32 can be represented as: 3 × 3 = 9 8 + 9 = 17 Thus, the given statement is incorrect. 65 can be written as 25 × 35. 65 = 25 × 35 (iv) 30 = (1000)0 Answer: True Explanation: Any number with the exponent 0 is considered as 1. 30 = 1 10000 = 1 Hence, the given expression is true. 1 = 1 30 = (1000)0 4. Express each of the following as a product of prime factors only in exponential form: (i) 108 × 192 Answer: 28 × 34 Explanation: 108 can be written as: = 2 × 54 = 2 × 2 × 27 = 2 × 2 × 3 × 9 = 2 × 2 × 3 × 3 × 3 = 22 × 33 192 can be written as: = 2 × 96 = 2 × 2 × 48 = 2 × 2 × 2 × 24 = 2 × 2 × 2 × 2 × 12 = 2 × 2 × 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 2 × 2 × 3 = 26 × 3 108 × 192 = (22 × 33) × (26 × 3) In the case of multiplication, the exponents are added if the base of numbers is the same. 108 × 192 = (22 + 6 × 33 + 1) 108 × 192 = 28 × 34 (ii) 270 Answer: 33 × 5 × 2 Explanation: 270 can be written as: = 27 × 10 = 3 × 3 × 3 × 5 × 2 = 33 × 5 × 2 (iii) 729 × 64 Answer: 36× 26 Explanation: 729 can be written as: = 3 × 243 = 3 × 3 × 81 = 3 × 3 × 3 × 27 = 3 × 3 × 3 × 3 × 9 = 3 × 3 × 3 × 3 × 3 × 3 = 36 64 can be written as: = 2 × 32 = 2 × 2 × 16 = 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 × 2 = 26 729 × 64 = 36× 26 (iv) 768 Answer: 28 × 3 Explanation: 768 can be written as: = 2 × 384 = 2 × 2 × 192 = 2 × 2 × 2 × 96 = 2 × 2 × 2 × 2 × 48 = 2 × 2 × 2 × 2 × 2 × 24 = 2 × 2 × 2 × 2 × 2 × 2 × 12 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 = 28 × 3 5. Simplify: (i) ((25)2 × 73)/ (83 × 7) Answer: 98 Explanation: To simplify, let's convert the available numbers into exponents. 8 can be written as: 2 × 2 × 2= 23 Substituting the value of 8 in the expression, ((25)2 × 73)/ (((23)3 × 7) = (210 × 73)/ (29 × 7) = (210 - 9 × 73 - 1) =(21 × 72) = 2 × 49 = 98 (ii) (25 × 52 × t8)/ (103 × t4) Answer: 5t4/8 Explanation: To simplify, let's convert the available numbers into exponents. 25 can be written as: 5 × 5 = 52 103 can be written as: 23 × 53 Substituting the value of 25 and 103 in the expression, (52 × 52 × t8)/ (23 × 53 × t4) = (52 + 2 - 3 × t8 - 4)/23 =(51 × t4)/23 = 5t4/8 (iii) (35 × 105 ×25)/ (57 × 65) Answer: 1 Explanation: To simplify, let's convert the available numbers into exponents. 25 can be written as: 5 × 5 = 52 105 can be written as: 25 × 55 65 can be written as: 25 × 35 Substituting the value of 25, 105, and 65 in the above expression, we get: (35 × 25 × 55 × 52)/ (57 × 25 × 35) = (35 - 5 × 25 - 5 × 55 + 2 - 7) = (30 × 20 × 50) = (1 × 1 × 1) = 1 Exercise 13.31. Write the following numbers in the expanded forms: 279404 279404 = 200000 + 70000 + 9000 + 400 + 4 Expanded form: = 2 × 105 + 7 × 104 + 9 × 103 + 4 × 102 + 0 × 101 + 4 × 100 3006194 3006194 = 3000000 + 6000 + 100 + 90 + 4 Expanded form: = 3 × 106 + 0 × 105 + 0 × 104 + 6 × 103 + 1 × 102 + 9 × 101 + 4 × 100 2806196 2806196 = 2000000 + 800000 + 6000 + 100 + 90 + 6 Expanded form: = 2 × 106 + 8 × 105 + 0 × 104 + 6 × 103 + 1 × 102 + 9 × 101 + 6 × 100 120719 120719 = 100000 + 20000 + 700 + 10 + 9 Expanded form: 1 × 105 + 2 × 104 + 0 × 103 + 7 × 102 + 1 × 101 + 9 × 100 20068 20068 = 20000 + 60 + 8 Expanded form: 2 × 104 + 0 × 103 + 0 × 102 + 6 × 101 + 8 × 100 2. Find the number from each of the following expanded forms: (a) 8 × 104 + 6 × 103 + 0 × 102 + 4 × 101 + 5 × 100 Answer: 86045 Explanation: 80000 + 6000 + 0 + 40 + 5 = 86045 (b) 4 × 105 + 5 × 103 + 3 × 102 + 2 × 100 Answer: 405302 Explanation: 400000 + 5000 + 300 + 2 = 405302 (c) 3 × 104 + 7 × 102 + 5 × 100 Answer: 30705 Explanation: 30000 + 700 + 5 30705 (d) 9 × 105 + 2 × 102 + 3 × 101 Answer: 900230 Explanation: 900000 + 200 + 30 900230 3. Express the following numbers in standard form: (i) 5,00,00,000 Answer: 5 × 107 Explanation: There are seven zeroes present in the number 5,00,00,000. Hence, it can be represented as: 5 × 107 (ii) 70,00,000 Answer: 7 × 106 Explanation: There are six zeroes present in the number 70,00,000. Hence, it can be represented as: 7 × 106 (iii) 3,18,65,00,000 Answer: 3.1865 × 109 Explanation: There are five zeroes present in the number 3,18,65,00,000. Hence, it can be represented as: 31865 × 105 Let's convert the number into the single decimal number. 3.1865 × 104 × 105 = 3.1865 × 109 (iv) 3,90,878 Answer: 3.90878 × 105 Explanation: 3,90,878 can be represented as: 3.90878 × 105 (v) 39087.8 Answer: 3.90878 × 104 Explanation: 39087.8 can be represented as: 3.90878 × 104 (vi) 3908.78 Answer: 3.90878 × 103 Explanation: 3908.78 can be represented as: 4. Express the number appearing in the following statements in standard form. (a) The distance between Earth and Moon is 384,000,000 m. Answer: 3.84 × 108 m Explanation: 384,000,000 m can be represented as: 384 × 106 = 3.84 × 102 × 106 = 3.84 × 108 m (b) Speed of light in vacuum is 300,000,000 m/s. Answer: 3 × 108 m/s Explanation: 300,000,000 m can be represented as: 3 × 108 m/s (c) Diameter of the Earth is 1,27,56,000 m. Answer: 1.2756 × 107 m Explanation: 1,27,56,000m can be represented as: 1.2756 × 107 m (d) Diameter of the Sun is 1,400,000,000 m. Answer: 1.4 × 109 m Explanation: 1,400,000,000 m can be represented as: 1.4 × 109 m (e) In a galaxy there are on an average 100,000,000,000 stars. Answer: 1 × 1011 Explanation: 100,000,000,000 stars can be represented as: 1011 Or 1 × 1011 (f) The universe is estimated to be about 12,000,000,000 years old. Answer: 1.2 × 1010 years Explanation: 12,000,000,000 can be represented as: 1.2 × 1010 years (g) The distance of the Sun from the centre of the Milky Way Galaxy is estimated to be 300,000,000,000,000,000,000 m. Answer: 3 × 1020 m Explanation: There are twelve zeroes in the number 300,000,000,000,000,000,000. Hence, it can be represented as: 3 × 1020 m (h) 60,230,000,000,000,000,000,000 molecules are contained in a drop of water weighing 1.8 gm. Answer: 6.023 × 1022 Explanation: 60,230,000,000,000,000,000,000 molecules can be represented as: 6.023 × 1022 molecules (i) The earth has 1,353,000,000 cubic km of sea water. Answer: 1.353 × 109 km3 Explanation: 1,353,000,000 cubic km can be represented as: 1.353 × 109 cubic km (j) The population of India was about 1,027,000,000 in March, 2001. Answer: 1.027 × 109 Explanation: 1,027,000,000 can be represented as: 1.027 × 109 Next TopicNCERT Solutions for class 8 Maths |