Exponents and Powers - Solutions 1

CBSE Class –VII Mathematics
NCERT Solutions
Chapter 13 Exponents and Powers (Ex. 13.1)

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Question 1. Find the value of:

(i) 26     (ii)93     (iii)112   (iv) 54

Answer: (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

Question 2.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 

Answer: (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 

Question 3. Express each of the following numbers using exponential notation:

(i) 512       (ii) 343          (iii) 729         (iv) 3125

Answer: (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

Question 4. Identify the greater number, wherever possible, in each of the following:

(i) 43 or 34

(ii) 53 or 35

(iii) 28 or 82

(iv) 1002 or 2100

(v) 210 or 102

Answer: (i) 43= 4 x 4 x 4 = 64

34= 3 x 3 x 3 x 3 = 81

Since 64 < 81

Thus, 34 is greater than 43.

(ii) 53$5^3$= 5 x 5 x 5 = 125

35$3^5$= 3 x 3 x 3 x 3 x 3 = 243

Since, 125 < 243

Thus, 34$3^4$ is greater than 53$5^3$.

(iii) 28$2^8$= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

82$8^2$= 8 x 8 = 64

Since, 256 > 64

Thus, 28$2^8$is greater than82$8^2$.

(iv) 1002${100}^2$= 100 x 100 = 10,000

2100$2^{100}$= 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 ……. X 2

Thus, 2100$2^{100}$is greater than1002${100}^2$.

(v) 210$2^{10}$= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1,024

102${10}^2$= 10 x 10 = 100

Since, 1,024 > 100

Thus, 210$2^{10}$> 102${10}^2$

Question 5. Express each of the following as product of powers of their prime factors:

(i) 648   (ii) 405    (iii) 540    (iv) 3,600

Answer: (i) 648 = 23×34$2^3\times 3^4$

(ii) 405 = 5×34$5\times 3^4$

(iii) 540 = 22×33×5$2^2\times 3^3\times 5$

(iv) 3,600 = 24×32×52$2^4\times 3^2\times 5^2$

Question 6. Simplify:

(i) 2×103$2\times {10}^3$

(ii) 72×22$7^2\times 2^2$

(iii)23×5$2^3\times 5$

(iv) 3×44$3\times 4^4$

(v)0×102$0\times {10}^2$

(vi) 52×33$5^2\times 3^3$

(vii)24×632$2^4\times {63}^2$

(viii) 32×104$3^2\times {10}^4$

Answer: (i) 2×103$2\times {10}^3$= 2 x 10 x 10 x 10 = 2,000

(ii) 72×22$7^2\times 2^2$= 7 x 7 x 2 x 2 = 196

(iii) 23×5$2^3\times 5$= 2 x 2 x 2 x 5 = 40

(iv) 3×44$3\times 4^4$= 3 x 4 x 4 x 4 x 4 = 768

(v) 0×102$0\times {10}^2$= 0 x 10 x 10 = 0

(vi) 52×33$5^2\times 3^3$= 5 x 5 x 3 x 3 x 3 = 675

(vii) 24×632$2^4\times {63}^2$= 2 x 2 x 2 x 2 x 3 x 3 = 144

(viii) 32×104$3^2\times {10}^4$= 3 x 3 x 10 x 10 x 10 x 10 = 90,000

Question 7. Simplify:

(i) (-4)3

(ii) (−3)×(−2)3$\left(-3\right)\times {\left(-2\right)}^3$

(iii) (−3)2×(−5)2${\left(-3\right)}^2\times {\left(-5\right)}^2$

(iv) (−2)3×(−10)3${\left(-2\right)}^3\times {\left(-10\right)}^3$

Answer: (i) (−4)3=(−4)×(−4)×(−4)=−64${\left(-4\right)}^3=\left(-4\right)\times \left(-4\right)\times \left(-4\right)=-64$

(ii) (−3)×(−2)3=(−3)×(−2)×(−2)×(−2)=24$\left(-3\right)\times {\left(-2\right)}^3=\left(-3\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)=24$

(iii) (−3)2×(−5)2=(−3)×(−3)×(−5)×(−5)=225${\left(-3\right)}^2\times {\left(-5\right)}^2=\left(-3\right)\times \left(-3\right)\times \left(-5\right)\times \left(-5\right)=225$

(iv) (−2)3×(−10)3=(−2)×(−2)×(−2)×(−10)×(−10)×(−10)=8000${\left(-2\right)}^3\times {\left(-10\right)}^3=\left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-10\right)\times \left(-10\right)\times \left(-10\right)=8000$

Question 8. Compare the following numbers:

(i) 2.7×1012$2.7\times {10}^{12}$;1.5 x 108$1.5x{10}^8$

(ii) 4×1014$4\times {10}^{14}$; 3×1017$3\times {10}^{17}$

Answer: (i) 2.7×1012$2.7\times {10}^{12}$and 1.5 x 108$1.5x{10}^8$

On comparing the exponents of base 10,

2.7×1012$2.7\times {10}^{12}$> 1.5 x 108$1.5x{10}^8$

(ii) 4×1014$4\times {10}^{14}$and 3×1017$3\times {10}^{17}$

On comparing the exponents of base 10,

4×1014$4\times {10}^{14}$< 3×1017