a,  8 2 . 32 4 =  2 3 2 . 2 5 4 = 2 3 . 2 . 2 5 . 4 = 2 6 . 2 20 = 2 6 + 20 = 2 26

b,  27 4 . 9 3 . 243 =  3 3 4 . 3 2 3 . 3 5 =  3 3 . 4 . 3 2 . 3 . 3 5 = 3 12 . 3 6 . 3 5 =  3 12 + 6 + 5 = 3 23

c,  13 2 - 12 2 = 169  - 144 = 25 =  5 2

d,  6 2 + 8 2 = 36 + 64 = 100 =  10 2

e,  5 3 + 5 4 + 125 2 . 5 3 =  5 3 + 5 4 + 5 3 . 2 . 5 3 =  5 3 + 5 4 + 5 6 . 5 3 =  5 3 . 1 + 5 + 5 3 . 5 3 = 5 6 . 131

f,  1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 1 + 8 + 27 + 64 + 125 = 225 =  15 2

\(2^{13}.4^5.8^2=2^{13}.\left(2^2\right)^5.\left(2^3\right)^2=2^{13}.2^{10}.2^6=2^{29}\)

\(2^{13}.4^5.8^2\)

\(=2^{13}.2^{10}.2^9\)

\(=2^{32}\)

2¹³.4⁵.8²=2¹³.(2²)⁵.(2³)²=2¹³.2¹⁰.2⁶=2^13+10+6=2²⁹

`@` `\text {Ans}`

`\downarrow`

`a)`

\(2^{13}\cdot4^5\cdot8^2\)

`=`\(2^{13}\cdot\left(2^2\right)^5\cdot\left(2^3\right)^2\)

`=`\(2^{13}\cdot2^{10}\cdot2^6\)

`=`\(2^{29}\)

`b)`

\(100^6. 1000^4. 1000000\)^\(33\)

`=`\(10^{12}\cdot10^{12}\cdot10^{198}=10^{222}\)

`c)`

\(24^5. 6^5. 8^8\)

`=`\(2^{15}\cdot3^5\cdot2^5\cdot3^5\cdot2^{24}\)

`=`\(2^{15+5+24}\cdot3^{5+5}\)

`=`\(2^{44}\cdot3^{10}\) 

a, \(2^{13}\cdot4^5\cdot8^2=2^{13}\cdot2^{10}\cdot2^6=2^{19}.\)

b, \(100^6\cdot1000^4\cdot1000000^{33}=10^{12}\cdot10^{12}\cdot10^{198}=10^{222}.\)

c, \(24^5\cdot6^5\cdot8^8=2^{15}\cdot3^5\cdot2^5\cdot3^5\cdot2^{24}=2^{44}\cdot3^{10}.\)

D = 4 3 + 6 3 + 7 3 + 2 = 625 = 25 2

a) \(9^3\cdot3^2\)

\(=\left(3^2\right)^3\cdot3^2\)

\(=3^6\cdot3^2\)

\(=3^8\)

b) \(x^7\cdot x:x^4\)

\(=x^8:x^4\)

\(=x^4\)

c) \(7\cdot3^9+3^{10}+51\cdot3^8\)

\(=3^8\cdot\left(7\cdot3+3^2+51\right)\)

\(=3^8\cdot81\)

\(=3^8\cdot3^4\)

\(=3^{12}\)

d) \(5^{15}\cdot125^3\cdot625\)

\(=5^{15}\cdot5^9\cdot5^4\)

\(=5^{28}\)

a)  6 2010  :  6 10  =  6 2000

b) ( 3 8  .  3 16  ) : ( 3 7  .  3 14 ) =  3 24  :  3 21  =  3 3

c) ( 2 26  :  2 10  ) : ( 2 18 :  2 16  ) =  2 16  : 2 2  =  2 14

d)  25 3  : 125 = ( 25 . 25 . 25 ) : 5 3  =  5 6  :  5 3  =  5 3

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220  x 100 = 22 000

=2²⁰.2².5²=2²².5²

`2^20 xx 100 = 2^20 xx 10^2 = 2^20 xx (2xx5)^2=2^22 xx 5^2`

243 : 3⁴ = 3⁵ : 3⁴ = 3^{5 - 4} = 3¹

3^1

\(243:3^4=3^5:3^4=3\)