Q1: Find the value of:
(i) 2^{6 }
(ii) 9^{3 }
(iii) 11^{2 }
(iv) 5^{4 }
Ans:
(i) 2^{6} = 2 x 2 x 2 x 2 x 2 x 2 = 64
(ii) 9^{3} = 9 x 9 x 9 = 729
(iii) 11^{2 }= 11 x 11= 121
(iv) 5^{4} = 5 x 5 x 5 x 5 = 625
Q2: Express the following in exponential form:
(i) 6 x 6 x 6 x 6
(ii) t x t
(iii) b x b x b x b
(iv) 5 x 5 x 7 x 7 x 7
(v) 2 x 2 x a x a
(vi) a x a x a x c x c x c x c x d
Ans:
(i) 6 x 6 x 6 x 6 = 6^{4}
(ii) t x t = t^{2}
(iii) b x b x b x b = b^{4}
(iv) 5 x 5 x 7 x 7 x 7 = 5^{2} x 7^{3}
(v) 2 x 2 x a x a = 2^{2 }x a^{2}
(vi) a x a x a x c x c x c x c x d = a^{3} x c^{4} x d
Q3: Express each of the following numbers using exponential notation:
(i) 512
(ii) 343
(iii) 729
(iv) 3125
Ans:
(i) 512
512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2^{9}
2 512  
2  256 
2  128 
2  64 
2  32 
2  16 
2  8 
2  4 
2  2 
 1 
(ii) 343
343 = 7 x 7 x 7 = 7^{3}
7 343  
7  49 
7  7 
 1 
(iii) 729
729 = 3 x 3 x 3 x 3 x 3 x 3 = 3^{6}
3 729  
3  243 
3  81 
3  27 
3  9 
3  3 
 1 
(iv) 3125
3125 = 5 x 5 x 5 x 5 x 5
5  3125 
5  625 
5  125 
5  25 
5  5 
 1 
Q4: Identify the greater number, wherever possible, in each of the following:
(i) 4^{3} and^{ 34}
(ii) 5^{3} or 3^{5}
(iii) 2^{8} or 8^{2}
(iv) 100^{2 }or 2^{100}
(v) 2^{10} or 10^{2}
Ans:
(i) 4^{3} = 4 x 4 x 4 = 64
3^{4} = 3 x 3 x 3 x 3 = 81
Since 64 < 81
Thus, 3^{4} is greater than 4^{3}.
(ii) 5^{3} = 5 x 5 x 5 = 125
3^{5} = 3 x 3 x 3 x 3 x 3 = 243
Since, 125 < 243
Thus, 3^{4} is greater than 5^{3}.
(iii) 2^{8}= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256
8^{2} = 8x8 = 64
Since, 256 > 64
Thus, 2^{8} is greater than 8^{2}.
(iv) 100^{2} = 100 x 100 = 10,000
2^{100} = 2 x 2 x 2 x 2 x 2 x.......14 times x....x 2 = 16,384 x....x 2
Since, 10,000 < 16,384 x....x2
Thus, 2^{100} is greater than 100^{2}.
(v) 2^{10} = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1,024
10^{2} = 10x10 = 100
Since, 1,024 > 100
Thus, 2^{10} >10^{2}
Q5: Express each of the following as product of powers of their prime factors:
(i) 648
(ii) 405
(iii) 540
(iv) 3,600
Ans:
(i) 648 = 2^{3} x 3^{4}
2  648 
2  324 
2  162 
3  81 
3  27 
3  9 
3  3 
 1 
(ii) 405 = 5 x 3^{4}
5  405 
3  81 
3  27 
3  9 
3  3 
 1 
(iii) 540 = 2^{2} x 3^{3} x 5
2  540 
2  270 
3  135 
3  45 
3  15 
5  5 
 1 
(iv) 3,600 = 2^{4} x 3^{2 }x 5^{2}
2  3600 
2  1800 
2  900 
2  450 
3  225 
3  75 
5  25 
5  5 
1 
Q6: Simplify:
(i) 2 x 10^{3}
(ii) 7^{2} x 2^{2}
(iii) 2^{3} x 5
(iv) 3 x 4^{4}
(v) 0 x 10^{2}
(vi) 5^{2 }x 3^{3}
(vii) 2^{4} x 3^{2}
(viii) 3^{2} x 10^{4}
Ans:
(i) 2 x 10^{3} = 2 x 10 x 10 x 10 = 2,000
(ii) 7^{2} x 2^{2} = 7 x 7 x 2 x 2 = 196
(iii) 2^{3} x 5 = 2 x 2 x 2 x 5 = 40
(iv) 3 x 4^{4} = 3 x 4 x 4 x 4 x 4 = 768
(v) 0 x 10^{2} = 0 x 10 x 10 = 0
(vi) 5^{3} x 3^{3} = 5 x 5 x 3 x 3 x 3 = 675
(vii) 2^{4 }x 3^{2} = 2 x 2 x 2 x 2 x 3 x 3 = 144
(viii) 3^{2} x 10^{4} = 3 x 3x 10x10x 10x10 = 90,000
Q7: Simplify:
(i) (4)^{3}
(ii) (3)x(2)^{3}
(iii) (3)^{2 }x (5)^{2}
(iv) (2)^{3} x (10)^{3}
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)
Question 8: Compare the following numbers:
(i) 2.7 x 10^{12}; 1.5 x 10^{8}
(ii) 4 x 10^{14}; 3 x 10^{17}
Answer 8:
(i) 2.7 x 10^{12 }and 1.5 x 10^{8}
On comparing the exponents of base 10,
2.7 x 10^{12} > 1.5x10^{8}
(ii) 4 x 10^{14} and 3 x 10^{17}
On comparing the exponents of base 10,
4 x 10^{14} < 3 x 10^{17}
Question 1: Using laws of exponents, simplify and write the answer in exponential form:
(i) 3^{2} x 3^{4} x 3^{8}
(ii) 6^{15} ÷ 6^{10}
(iii) a^{3} x a^{2 }
(iv) 7^{x} x 7^{2}
(v) (5^{2})^{2} ÷ 5^{3}
(vi) 2^{5} x 5^{5}
(vii) a^{4} x b^{4}
(viii) (3^{4})^{3}
(ix) [2^{20} ÷ 2^{15}) x 2^{3}
(x) 8^{t} x 8^{2}
Answer 1:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Q2: Simplify and express each of the following in exponential form:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
Ans:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
Q3: Say true or false and justify your answer:
(i) 10 x 10^{11 }= 10011
(ii) 2^{3} > 5^{2}
(iii) 2^{3}x 3^{2} = 6s
(iv) 3^{0} = (1000)^{0}
Ans:
(i) 10x10^{11} = 100^{11}
L.H.S. 10^{1+11} = 10^{12 } and R.H.S. (10^{2})^{11}=10^{22}
Since, L.H.S. ≠ R.H.S.
Therefore, it is false.
(ii) 2^{3} > 5^{2}
L.H.S 2^{3} = 8 and R.H.S. 5^{2} = 25
Since, L.H,S. is not greater than R.H.S.
Therefore, it is false,
(iii) 2^{3} x3^{2} = 6^{5}
L.H.S. 2^{3} x 3^{2} = 8 x 9 = 72 and R.H.S. 6^{5} =7.776
Since, L.H.S. ≠ R.H.S.
Therefore, it is false.
(iv) 3^{0}= (1000)^{0}
L.H.S. 3^{0} = 1 and R.H.S. (1000)^{0} = 1
Since, L.H.S. = R.H.S.
Therefore, it is true.
Q4: Express each of the following as a product of prime factors only in exponential form:
(i) 108 x 192
(ii) 270
(iii) 729 x 64
(iv) 768
Ans:
Question 5: Simplify:
(i)
(ii)
(iii)
Answer 5:
(i)
(ii)
(iii)
279404,
3006194,
2806196,
120719,
20068
Answer 1:
(i) 2,79,404
= 2,00,000 + 70,000 + 9,000 + 400 + 00 + 4
= 2x100000 + 7x 10000+ 9x1000 + 4 x 100+ 0x 10 + 4x1
= 2x10^{5}+7x10^{4} +9x10^{3} + 4 x10^{2}+ 0x10^{1}+4x10^{0}
(ii) 30,06,194
= 30,00,000 + 0 + 0 + 6,000 + 100 + 90 + 4
= 3 x 1000000 + 0 x 100000 + 0 x 10000 + 6 x 1000 + 1x100 + 9x10 + 4x1
= 3x10^{6} + 0x10^{5} +0x10^{4} +6 x10^{3}+1x10^{2} +9x10+4x10^{0}
(iii) 28,06,196
= 20,00,000 + 8,00,000 + 0 + 6,000 + 100 + 90 + 6
= 2x1000000 + 8x 100000+ 0x 10000 +6 x 1000 + 1 x 100 + 9 x 10 + 6 x 1
= 2 x10^{6}+8x10^{5}+0x10^{4} + 6x10^{3}+1x10^{2}+9x10+6x10^{0}
(iv) 1,20,719
= 1,00,000 + 20,000 + 0 + 700 + 10 + 9
= 1 x 100000 + 2 x 10000 + 0x 1000 + 7x100 + 1x 10+9x1
= 1 x 10^{5} + 2 x 10^{4 }+ 0 x 10^{3} + 7 x 10^{2} + 1 x 10^{1} + 9 x 10^{0}
(v) 20,068
= 20,000 + 00 + 00 + 60 + 8
= 2 x 10000 + 0 x 1000 + 0 x 100 + 6 x 10 + 8 x 1
Question 2:
Find the number from each of the following expanded forms:
(a) 8 x 104 + 6 x 103 + 0 x 102 + 4 x 101 + 5 x 100
(b) 4 x 105 + 5 x 103 + 3 x 102 + 2 x 100
(c) 3 x 104 + 7 x 102 + 5 x 100 (d) 9 x 105 + 2 x 102 + 3 x 101
Answer 2:
(a) 8 x 104 + 6 x 10^{3} + 0 x 10^{2} + 4 x 10^{1} + 5 x 10^{0}
= 8 x 10000 + 6x 1000 + 0 x 100 + 4 x 10 + 5 x 1
= 80000 + 6000 + 0 + 40+5
= 86,045
(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 + 3000 + 0 + 2
= 4,05,302
(c) 3 x 10^{4} + 7 x 10z + 5 x 10^{0}
= 3 x 10000 + 0 x 1000 + 7 x 100 + 0 x 10 + 5 x 1
= 30000 + 0 + 700 + 0 + 5
= 30,705
(d) 9x 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
= 9,00,230
Question 3:
Express the following numbers in standard form:
(i) 5,00,00,000
(ii) 70,00,000
(iii) 3,18,65,00,000
(iv) 3,90,878
(v) 39087.8
(vi) 3908.78
Answer 3:
(i) 5,00,00,000 = 5 x 1,00,00,000 = 5 x 10^{7}
(ii) 70,00,000 = 7 x 10,00,000 = 7 x 10^{6}
(iii) 3,18,65,00,000 = 31865 x 100000
= 3.1865 x 10000 x 100000 = 3.1865 x 10^{9}
(iv) 3,90,878 = 3.90878 x 100000 = 3.90878 x 10^{5}
(v) 39087.8 = 3.90878 x 10000 = 3.90878 x 10^{4}
(vi) 3908.78 = 3.90878 x 1000 = 3.90878 x 10^{3}
Question 4:
Express the number appearing in the following statements in standard form:
(a) The distance between Earth and Moon is 384,000,000 m.
(b) Speed of light in vacuum is 300,000,000 m/s.
(c) Diameter of Earth id 1,27,56,000 m.
(d) Diameter of the Sun is 1,400,000,000 m.
(e) In a galaxy there are on an average 100,000,000,0000 stars.
(f) The universe is estimated to be about 12,000,000,000 years old.
(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.
(h) 60,230,000,000,000,000,000,000 molecules are contained in a drop of water weighing 1.8 gm.
(i) The Earth has 1,353,000,000 cubic km of sea water.
(j) The population of India was about 1,027,000,000 in march, 2001.
Answer 4:
(a) The distance between Earth and Moon = 384,000,000 m
= 384x 1000000 m
= 3.84 x 100 x 1000000
= 3.84x10^{8} m
(b) Speed of light in vacuum = 300,000,000 m/s
= 3 x 100000000 m/s
= 3x10^{8} m/s
(c) Diameter of the Earth = 1,27,50,000 m
= 12756 x 1000 m
= 1.2756 x 10000 x 1000 m
= 1.2756x10^{7} m
(d) Diameter of the Sun = 1,400,000,000 m
= 14 x 100,000,000 m
= 1.4 x 10 x 100,000,000 m
= 1.4 x 10^{9} m
(e) Average of Stars = 100, 000, 000,000
= 1 x 100,000,000,000
= 1x10^{11}
(f) Years of Universe = 12,000,000,000 years
= 12 x 1000,000,000 years
= 1.2 x 10 x 1000,000,000 years
= 1.2 x 10^{10} years
(g) Distance of tlie Sun from the centre of the Milky Way Galaxy
= 300,000,000,000,000,000,000 m
= 3 x 100,000,000,000,000,000,000 m
= 3 x 10^{20} m
(h) Number of molecules in a drop of water weighing 1.8 gm
= 60,230,000,000,000,000,000,000
= 6023 x 10,000,000,000,000,000,000
= 6,023 x 1000 x 10,000,000,000,000,000,000
= 6.023 x10^{22}
(i) The Earth has Sea water = 1,353,000,000 km^{3}
= 1,353 x 1000000 km^{3}
= 1,353 x 1000 x 1000,000 km^{3}
= 1.353x10^{9} km^{3}
(j) The population of India = 1,027,000,000
= 1027x1000000 = 1,027x1000x1000000
= 1.027x10^{9}
76 videos345 docs39 tests

1. What is the expanded form of a number? 
2. How do you write a number in expanded form? 
3. Why is it important to understand expanded form? 
4. Can you convert any number into expanded form? 
5. How can expanded form help in simplifying mathematical calculations? 

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