**Exercise 13.1**

Question 1:

Find the value of:

(i) 2

(iii) 11

(i) 2

^{6}(ii) 9^{3}(iii) 11

^{2}(iv)5^{4}^{Answer: }(i) 2

(ii) 9

(iii) 11

(iv)5

Question 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 = 625Question 2:

Express the following in exponential form:

(i) 6 × 6 × 6 × 6 (ii)

(iii)

(v) 2 × 2 ×

(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**Answer:*(i) 6 × 6 × 6 × 6 = 6

(ii)

(iii)

(iv) 5 × 5 × 7 × 7 × 7 = 5

(v) 2 × 2 ×

(vi)

Question 3:

^{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 the following numbers using exponential notation:

(i) 512 (ii) 343

(iii) 729 (iv) 3125

Answer:

(i) 512 (ii) 343

(iii) 729 (iv) 3125

Answer:

(i) 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2

(ii) 343 = 7 × 7 × 7 = 7

(iii) 729 = 3 × 3 × 3 × 3 × 3 × 3 = 3

(iv) 3125 = 5 × 5 × 5 × 5 × 5 = 5

^{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

(iii) 2

(v) 2

(i) 4

^{3}or 3^{4}(ii) 5^{3}or 3^{5}(iii) 2

^{8}or 8^{2}(iv) 100^{2 }or 2^{100}(v) 2

^{10}or 10^{2}^{Answer: }(i) 4

3

Therefore, 3

(ii) 5

3

Therefore, 3

(iii) 2

8

Therefore, 2

(iv)100

2

2

100

Therefore, 2

(v) 2

2

10

Therefore, 2

^{3}= 4 × 4 × 4 = 643

^{4}= 3 × 3 × 3 × 3 = 81Therefore, 3

^{4}> 4^{3}(ii) 5

^{3}= 5 × 5 × 5 =1253

^{5}= 3 × 3 × 3 × 3 × 3 = 243Therefore, 3

^{5}> 5^{3}(iii) 2

^{8}= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2568

^{2}= 8 × 8 = 64Therefore, 2

^{8}> 8^{2}(iv)100

^{2}or 2^{100}2

^{10}= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 10242

^{100}= 1024 × 1024 × 1024 × 1024 × 1024 × 1024 × 1024 × 1024 ×1024 × 1024100

^{2}= 100 × 100 = 10000Therefore, 2

^{100}> 100^{2}(v) 2

^{10}and 10^{2}2

^{10}= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 102410

^{2}= 10 × 10 = 100Therefore, 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) 3,600

(i) 648 (ii) 405

(iii) 540 (iv) 3,600

^{Answer: }(i) 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2

(ii) 405 = 3 × 3 × 3 × 3 × 5 = 3

(iii) 540 = 2 × 2 × 3 × 3 × 3 × 5 = 2

(iv) 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 = 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

(iii) 2

(v) 0 × 10

(vii) 2

(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) 3^{2}× 10^{4}^{Answer: }(i) 2 × 10

(ii) 7

(iii) 2

(iv) 3 × 4

(v) 0 × 10

(vi) 5

(vii) 2

(viii) 3

Question 7:

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

^{2}× 10^{4}= 3 × 3 × 10 × 10 × 10 × 10 = 9 × 10000 = 90000Question 7:

Simplify:

(i) (− 4)

(iii) (− 3)

(i) (− 4)

^{3}(ii) (− 3) × (− 2)^{3}(iii) (− 3)

^{2}× (− 5)^{2}(iv)(− 2)^{3}× (−10)^{3}^{Answer: }(i) (−4)

(ii) (−3) × (−2)

(iii) (−3)

(iv) (−2)

= (−8) × (−1000) = 8000

Question 8:

^{3}= (−4) × (−4) × (−4) = −64(ii) (−3) × (−2)

^{3}= (−3) × (−2) × (−2) × (−2) = 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

(ii) 4 × 10

(i) 2.7 × 10

^{12}; 1.5 × 10^{8}(ii) 4 × 10

^{14}; 3 × 10^{17}^{Answer: }(i) 2.7 × 10

2.7 × 10

(ii) 4 × 10

3 × 10

^{12}; 1.5 × 10^{8}2.7 × 10

^{12}> 1.5 × 10^{8}(ii) 4 × 10

^{14}; 3 × 10^{17}3 × 10

^{17}> 4 × 10^{14}**Exercise 13.2**^{ }^{Question 1:}Using laws of exponents, simplify and write the answer in exponential form:

(i) 3

(iv) 7

(vii)

(ix) (x) 8

(i) 3

^{2}× 3^{4}× 3^{8}(ii) 6^{15}÷ 6^{10}(iii)*a*^{3}×*a*^{2}(iv) 7

^{x}× 7^{2}(v) (vi) 2^{5}× 5^{5}(vii)

*a*^{4}×*b*^{4}(viii) (3^{4})^{3}(ix) (x) 8

^{t}÷ 8^{2}^{Answer:}(i) 3

= 3

(ii) 6

= 6

(iii)

=

(iv) 7

(v) (5

= 5

= 5

= 5

= 5

(vi) 2

= (2 × 5)

= 10

(vii)

= (

(viii) (3

(ix) (2

= (2

= 2

= (2

= 2

(x) 8

Question 2:

^{2}× 3^{4}× 3^{8}= (3)^{2 + 4 + 8}(*a*^{m}×*a*^{n}=*a*^{m}^{+}^{n})= 3

^{14}(ii) 6

^{15}÷ 6^{10}= (6)^{15 − 10}(*a*^{m}÷*a*^{n}=*a*^{m}^{−}^{n})= 6

^{5}(iii)

*a*^{3}×*a*^{2 }=*a*^{(3 + 2) }(*a*^{m}×*a*^{n}=*a*^{m}^{+}^{n})=

*a*^{5}(iv) 7

^{x}+ 7^{2}= 7^{x}^{ + 2}(*a*^{m}×*a*^{n}=*a*^{m}^{+}^{n})(v) (5

^{2})^{3}÷ 5^{3}= 5

^{2 × 3}÷ 5^{3 }(*a*^{m})^{n}=*a*^{mn}= 5

^{6}÷ 5^{3}= 5

^{(6 − 3) }(*a*^{m}÷*a*^{n}=*a*^{m}^{−}^{n})= 5

^{3}(vi) 2

^{5}× 5^{5}= (2 × 5)

^{5}[*a*^{m}×*b*^{m}= (a ×*b*)^{m}]= 10

^{5}(vii)

*a*^{4}×*b*^{4}= (

*ab*)^{4}[*a*^{m}×*b*^{m}= (a ×*b*)^{m}](viii) (3

^{4})^{3}= 3^{4 × 3}= 3^{12}(*a*^{m})^{n}=*a*^{mn}(ix) (2

^{20}÷ 2^{15}) × 2^{3}= (2

^{20 − 15})^{ }× 2^{3}(*a*^{m}÷*a*^{n}=*a*^{m}^{−}^{n})= 2

^{5}× 2^{3}= (2

^{5 + 3}) (*a*^{m}×*a*^{n}=*a*^{m}^{+}^{n})= 2

^{8}(x) 8

^{t}÷ 8^{2}= 8^{(}^{t}^{ − 2)}(*a*^{m}÷*a*^{n}=*a*^{m}^{−}^{n})Question 2:

Simplify and express each of the following in exponential form:

(i) (ii) (iii)

(iv) (v) (vi) 2

(vii) 2

(x) (xi) (xii)

(i) (ii) (iii)

(iv) (v) (vi) 2

^{0}+ 3^{0}+ 4^{0}(vii) 2

^{0}× 3^{0}× 4^{0}(viii) (3^{0}+ 2^{0}) × 5^{0}(ix)(x) (xi) (xii)

^{Answer: }(i)

(ii) [(5

= [5

= [5

= [5

= 5

= 5

= 5

(iii) 25

= (5

= 5

= 5

= 5

= 5

(iv)

= 1 × 7 × 11

(v)

(vi) 2

(vii) 2

(viii) (3

(ix)

(x)

(xi)

(xii) (2

= (2

= 2

(ii) [(5

^{2})^{3}× 5^{4}] ÷ 5^{7}= [5

^{2 × 3}× 5^{4}] ÷ 5^{7}(*a*^{m})^{n}=*a*^{mn}= [5

^{6}× 5^{4}] ÷ 5^{7}= [5

^{6 + 4}] ÷ 5^{7}(*a*^{m}×*a*^{n}=*a*^{m}^{+}^{n})= 5

^{10}÷ 5^{7}= 5

^{10 − 7}(*a*^{m}÷*a*^{n}=*a*^{m}^{−}^{n})= 5

^{3}(iii) 25

^{4}÷ 5^{3}= (5 ×5)^{4}÷ 5^{3}= (5

^{2})^{4}÷ 5^{3}= 5

^{2 × 4}÷ 5^{3}(*a*^{m})^{n}=*a*^{mn}= 5

^{8}÷ 5^{3}= 5

^{8 − 3}(*a*^{m}÷*a*^{n}=*a*^{m}^{−}^{n})= 5

^{5}(iv)

= 1 × 7 × 11

^{5}= 7 × 11^{5}(v)

(vi) 2

^{0}+ 3^{0}+ 4^{0}= 1 + 1 + 1 = 3(vii) 2

^{0}× 3^{0}× 4^{0}= 1 × 1 × 1 = 1(viii) (3

^{0}+ 2^{0}) × 5^{0}= (1 + 1) × 1 = 2(ix)

(x)

(xi)

(xii) (2

^{3}× 2)^{2}=^{ }(*a*^{m}×*a*^{n}=*a*^{m}^{+}^{n})= (2

^{4})^{2}= 2^{4 × 2}(*a*^{m})^{n}=*a*^{mn}= 2

^{8}^{ }^{ Question 3:}Say true or false and justify your answer:

(i) 10 × 10

(iii) 2

(i) 10 × 10

^{11}= 100^{11}(ii) 2^{3}> 5^{2}(iii) 2

^{3}× 3^{2}= 6^{5 }(iv) 3^{0}= (1000)^{0}^{Answer: }(i) 10 × 10

L.H.S. = 10 × 10

= 10

R.H.S. = 100

= 10

As L.H.S. ≠ R.H.S.,

Therefore, the given statement is false.

(ii) 2

L.H.S. = 2

R.H.S. = 5

As 25 > 8,

Therefore, the given statement is false.

(iii) 2

L.H.S. = 2

R.H.S. = 6

As L.H.S. ≠ R.H.S.,

Therefore, the given statement is false.

(iv) 3

L.H.S. = 3

R.H.S. = (1000)

Therefore, the given statement is true.

Question 4:

^{11}= 100^{11}L.H.S. = 10 × 10

^{11}= 10^{11 + 1 }(*a*^{m}×*a*^{n}=*a*^{m}^{+}^{n})= 10

^{12}R.H.S. = 100

^{11}= (10 ×10)^{11}= (10^{2})^{11}= 10

^{2 × 11}= 10^{22}(*a*^{m})^{n}=*a*^{mn}As L.H.S. ≠ R.H.S.,

Therefore, the given statement is false.

(ii) 2

^{3}> 5^{2}L.H.S. = 2

^{3}= 2 × 2 × 2 = 8R.H.S. = 5

^{2}= 5 × 5 = 25As 25 > 8,

Therefore, the given statement is false.

(iii) 2

^{3}× 3^{2}= 6^{5}L.H.S. = 2

^{3}× 3^{2}= 2 × 2 × 2 × 3 × 3 = 72R.H.S. = 6

^{5}= 7776As L.H.S. ≠ R.H.S.,

Therefore, the given statement is false.

(iv) 3

^{0}= (1000)^{0}L.H.S. = 3

^{0}= 1R.H.S. = (1000)

^{0}= 1 = L.H.S.Therefore, the given statement is true.

Question 4:

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

(i) 108 × 192 (ii) 270

(iii) 729 × 64

Answer:

(i) 108 × 192 (ii) 270

(iii) 729 × 64

^{ }(iv) 768Answer:

(i) 108 × 192

= (2 × 2 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2 × 3)

= (2

= 2

= 2

(ii) 270 = 2 × 3 × 3 × 3 × 5 = 2 × 3

(iii) 729 × 64 = (3 × 3 × 3 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2)

= 3

(iv) 768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 = 2

Question 5:

= (2 × 2 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2 × 3)

= (2

^{2}× 3^{3}) × (2^{6}× 3)^{ }= 2

^{6 + 2}× 3^{3 + 1}(*a*^{m}×*a*^{n}=*a*^{m}^{+}^{n})= 2

^{8}× 3^{4}(ii) 270 = 2 × 3 × 3 × 3 × 5 = 2 × 3

^{3 }× 5(iii) 729 × 64 = (3 × 3 × 3 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2)

= 3

^{6}× 2^{6}(iv) 768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 = 2

^{8}× 3Question 5:

Simplify:

(i) (ii) (iii)

(i) (ii) (iii)

^{ Answer:}(i)

(ii)

(iii)

(ii)

(iii)

^{ }**Exercise 13.3**^{ }^{Question 1:} Write the following numbers in the expanded forms:

279404, 3006194, 2806196, 120719, 20068

Answer:

279404, 3006194, 2806196, 120719, 20068

Answer:

279404 = 2 × 10

3006194 = 3 × 10

2806196 = 2 × 10

120719 = 1 × 10

20068 = 2 × 10

^{5}+ 7 × 10^{4}+ 9 × 10^{3}+ 4 × 10^{2}+ 0 × 10^{1}+ 4 × 10^{0}3006194 = 3 × 10

^{6}+ 0 × 10^{5}+ 0 × 10^{4}+ 6 × 10^{3}+ 1 × 10^{2}+ 9 × 10^{1}+ 4 × 10^{0}2806196 = 2 × 10

^{6}+ 8 × 10^{5}+ 0 × 10^{4}+ 6 × 10^{3}+ 1 × 10^{2}+ 9 × 10^{1}+ 6 × 10^{0}120719 = 1 × 10

^{5}+ 2 × 10^{4}+ 0 × 10^{3}+ 7 × 10^{2}+ 1 × 10^{1}+ 9 × 10^{0}20068 = 2 × 10

^{4}+ 0 × 10^{3}+ 0 × 10^{2}+ 6 × 10^{1}+ 8 × 10^{0}^{Question 2:} Find the number from each of the following expanded forms:

(a) 8 × 10

(b) 4 × 10

(c) 3 × 10

(d) 9 × 10

(a) 8 × 10

^{4}+ 6 × 10^{3}+ 0 × 10^{2}+ 4 × 10^{1}+ 5 × 10^{0}(b) 4 × 10

^{5 }+ 5 × 10^{3 }+ 3 × 10^{2 }+ 2 × 10^{0}(c) 3 × 10

^{4}+ 7 × 10^{2}+ 5 × 10^{0}(d) 9 × 10

^{5}+ 2 × 10^{2}+ 3 × 10^{1}^{Answer: } (a) 8 × 10

= 86045

(b) 4 × 10

= 405302

(c) 3 × 10

= 30705

(d) 9 × 10

= 900230

Question 3:

^{4}+ 6 × 10^{3}+ 0 × 10^{2}+ 4 × 10^{1}+ 5 × 10^{0}= 86045

(b) 4 × 10

^{5}+ 5 × 10^{3}+ 3 × 10^{2}+ 2 × 10^{0}= 405302

(c) 3 × 10

^{4}+ 7 × 10^{2}+ 5 × 10^{0}= 30705

(d) 9 × 10

^{5}+ 2 × 10^{2}+ 3 × 10^{1}= 900230

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:

(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:

(i) 50000000 = 5 × 10

(ii) 7000000 = 7 × 10

(iii) 3186500000 = 3.1865 × 10

(iv) 390878 = 3.90878 × 10

(v) 39087.8 = 3.90878 × 10

(vi) 3908.78 = 3.90878 × 10

^{7}(ii) 7000000 = 7 × 10

^{6}(iii) 3186500000 = 3.1865 × 10

^{9}(iv) 390878 = 3.90878 × 10

^{5}(v) 39087.8 = 3.90878 × 10

^{4}(vi) 3908.78 = 3.90878 × 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 the Earth is 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, 000 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:

(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 the Earth is 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, 000 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:

(a) 3.84 × 10

(b) 3 × 10

(c) 1.2756 × 10

(d) 1.4 × 10

(e) 1 × 10

(f) 1.2 × 10

(g) 3 × 10

(h) 6.023 × 10

(i) 1.353 × 10

(j) 1.027 × 10

^{8}m(b) 3 × 10

^{8}m/s(c) 1.2756 × 10

^{7}m(d) 1.4 × 10

^{9}m(e) 1 × 10

^{11}stars(f) 1.2 × 10

^{10}years(g) 3 × 10

^{20}m(h) 6.023 × 10

^{22}(i) 1.353 × 10

^{9}cubic km(j) 1.027 × 10

^{9}^{ }^{ }^{ }^{ }^{ }^{ }^{ }^{ }^{ }^{ }^{ }

thanks for this stuff!

ReplyDeletepls give questions(ncert) based on congruence of triangles also...

ReplyDeleteThanks for the help to enhance results of exams....thanks Aman Bhatia

ReplyDelete