chapter 11

(i) $2^6$

(ii) $9^3$

(iii) $11^2$

(iv) $5^4$

(i) $6 \times 6 \times 6 \times 6$ $\implies$ $\mathbf{6^4}$

(ii) $t \times t$ $\implies$ $\mathbf{t^2}$

(iii) $b \times b \times b \times b$ $\implies$ $\mathbf{b^4}$

(iv) $5 \times 5 \times 7 \times 7 \times 7$ $\implies$ $\mathbf{5^2 \times 7^3}$

(v) $2 \times 2 \times a \times a$ $\implies$ $\mathbf{2^2 \times a^2}$

(vi) $a \times a \times a \times c \times c \times c \times c \times d$ $\implies$ $\mathbf{a^3 \times c^4 \times d}$

(i) $512$

(ii) $343$

(iii) $729$

(અથવા $9^3$)

(iv) $3125$

(i) $4^3$ અને $3^4$

• 
  

  $4^3 = 4 \times 4 \times 4 = 64$

  
  


• 
  

  $3^4 = 3 \times 3 \times 3 \times 3 = 81$

  
  


• 
  

  અહીં, $81 > 64$ હોવાથી મોટી સંખ્યા ન્હ $3^4$ છે.

  
  

(ii) $5^3$ અને $3^5$

• 
  

  $5^3 = 5 \times 5 \times 5 = 125$

  
  


• 
  

  $3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$

  
  


• 
  

  મોટી સંખ્યા ન્હ $3^5$ છે.

  
  

(iii) $2^8$ અને $8^2$

• 
  

  $2^8 = 256$

  
  


• 
  

  $8^2 = 64$

  
  


• 
  

  મોટી સંખ્યા ન્હ $2^8$ છે.

  
  

(iv) $100^2$ અને $2^{100}$

• 
  

  $100^2 = (10^2)^2 = 10^4 = 10,000$

  
  


• 
  

  $2^{100}$ એ ખૂબ જ મોટી સંખ્યા છે ($10^4$ કરતાં અનેકગણી મોટી).

  
  


• 
  

  મોટી સંખ્યા ન્હ $2^{100}$ છે.

  
  

(v) $2^{10}$ અને $10^2$

• 
  

  $2^{10} = 1024$

  
  


• 
  

  $10^2 = 100$

  
  


• 
  

  મોટી સંખ્યા ન્હ $2^{10}$ છે.

  
  

(i) $648$

(ii) $405$

(iii) $540$

(iv) $3,600$

(i) 2×103

(ii) 72×22

(iii) 23×5

(iv) 3×44

(v) 0×102

(vi) 52×33

(vii) 24×32

(viii) 32×104

(i) $(-4)^3$

(ii) $(-3) \times (-2)^3$

(iii) $(-3)^2 \times (-5)^2$

(iv) $(-2)^3 \times (-10)^3$

(i) $2.7 \times 10^{12}$; $1.5 \times 10^8$

• 
  

  બંને સંખ્યાઓમાં $10$ ની ઘાત જોઈએ તો $10^{12}$ એ $10^8$ કરતાં મોટી છે.

  
  


• 
  

  તેથી, $2.7 \times 10^{12} > 1.5 \times 10^8$

  
  

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

• 
  

  અહીં $10^{17}$ વાળી સંખ્યા મોટી છે ($10^{17} > 10^{14}$).

  
  


• 
  

  તેથી, $3 \times 10^{17} > 4 \times 10^{14}$

  
  

(i) $3^2 \times 3^4 \times 3^8$

(ii) $6^{15} \div 6^{10}$

(iii) $a^3 \times a^2$

(iv) $7^x \times 7^2$

(v) $(5^2)^3 \div 5^3$

(vi) $2^5 \times 5^5$

(vii) $a^4 \times b^4$

(viii) $(3^4)^3$

(ix) $(2^{20} \div 2^{15}) \times 2^3$

(x) $8^t \div 8^2$

(i) 3×3223×34×4​

(ii) [(52)3×54]÷57

(iii) 254÷53

(iv) 21×1133×72×118​

(v) 34×3337​

(vi) 20+30+40

(vii) 20×30×40

(viii) (30+20)×50

(ix) 43×a328×a5​

(x) (a3a5​)×a8

(xi) 45×a5b245×a8b3​

(xii) (23×2)2

(i) 10×1011=10011

• 
  

  ડા.બા.: 101×1011=1012

  
  


• 
  

  જ.બા.: 10011=(102)11=1022

  
  


• 
  

  1012=1022⟹ ખોટું

  
  

(ii) 23>52

• 
  

  23=8 અને 52=25

  
  


• 
  

  8>25 શક્ય નથી ⟹ ખોટું

  
  

(iii) 23×32=65

• 
  

  ડા.બા.: 8×9=72

  
  


• 
  

  જ.બા.: 65=7776

  
  


• 
  

  72=7776⟹ ખોટું

  
  

(iv) 30=(1000)0

• 
  

  30=1 અને (1000)0=1

  
  


• 
  

  1=1⟹ ખરું

  
  

(i) 108×192

• 
  

  108=22×33

  
  


• 
  

  192=26×31

  
  


• 
  

  108×192=(22×33)×(26×31)=22+6×33+1=28×34

  
  

(ii) 270

• 
  

  270=2×3×3×3×5=2×33×5

  
  

(iii) 729×64

• 
  

  729=36

  
  


• 
  

  64=26

  
  


• 
  

  729×64=26×36

  
  

(iv) 768

• 
  

  768=2×2×2×2×2×2×2×2×3=28×3

  
  

(i) $108 \times 192$

• 
  

  $108 = 2^2 \times 3^3$

  
  


• 
  

  $192 = 2^6 \times 3^1$

  
  


• 
  

  $108 \times 192 = (2^2 \times 3^3) \times (2^6 \times 3^1) = 2^{2+6} \times 3^{3+1} = \mathbf{2^8 \times 3^4}$

  
  

(ii) $270$

• 
  

  $270 = 2 \times 3 \times 3 \times 3 \times 5 = \mathbf{2 \times 3^3 \times 5}$

  
  

(iii) $729 \times 64$

• 
  

  $729 = 3^6$

  
  


• 
  

  $64 = 2^6$

  
  


• 
  

  $729 \times 64 = \mathbf{2^6 \times 3^6}$

  
  

(iv) $768$

• 
  

  $768 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 = \mathbf{2^8 \times 3}$

  
  

(a) $279404$

(b) $3006194$

(c) $2806196$

(d) $120719$

(e) $20068$

(a) $8 \times 10^4 + 6 \times 10^3 + 0 \times 10^2 + 4 \times 10^1 + 5 \times 10^0$

(b) $4 \times 10^5 + 5 \times 10^3 + 3 \times 10^2 + 2 \times 10^0$

(c) $3 \times 10^4 + 7 \times 10^2 + 5 \times 10^0$

(d) $9 \times 10^5 + 2 \times 10^2 + 3 \times 10^1$

(i) $5,00,00,000$

(ii) $70,00,000$

(iii) $3,18,65,00,000$

(iv) $3,90,878$

(v) $39087.8$

(vi) $3908.78$

(i) $5,00,00,000$

(ii) $70,00,000$

(iii) $3,18,65,00,000$

(iv) $3,90,878$

(v) $39087.8$

(vi) $3908.78$