Hello

a)  4⁸.8⁴ 

     = (2²)⁸.(2³)⁴ = 2¹⁶.2¹² = 2²⁸

b)  4¹⁵6³⁰ 

     = (2.2)¹⁵.(2.3)³⁰ 

     = 2¹⁵.2¹⁵.2³⁰.3³⁰ 

     = 2⁶⁰.3³⁰ = 2^2.30.3³⁰ 

     = (2²)³⁰.(3)³⁰ 

     = (2².3)³⁰ 

     = (12)³⁰ =12³⁰

c)  2¹⁰.15 + 2¹⁰.85

     = 2¹⁰(15+85)

     =2¹⁰(100)

     = 2¹⁰(10)² 

     = 2¹⁰.(2.5)² 

     = 2¹⁰.2².5² 

     = 2¹².5² 

     = 2^6.2.5²

     = (2⁶)².5² 

     = (2⁶.5)² 

     = (64.5)² 

     = (320)² 

     = 320²

d)  5¹².7 - 5¹¹.10

     = 5¹².7 - 5¹¹.5.2 

     = 5¹².7 - 5¹².2

     =  5¹²(7 - 2)

     = 5¹².5 

     = 5¹³

e) x¹.x².x³....x¹⁰⁰

     = x⁵⁰⁵⁰     

\(\left(x+1\right)^3=10\Rightarrow x+1=\sqrt[3]{10}\Rightarrow x=\sqrt[3]{10}-1\)

3 mũ 2 x + 2 + 3 mũ 2x + 1 = 108

\(x^2-2x=378\Rightarrow x^2-2x-378=0\Rightarrow x^2-2x+1-377=0\Rightarrow\left(x-1\right)^2-377=0\Rightarrow\left(x-1-\sqrt{377}\right)\left(x-1+\sqrt{377}\right)=0\Rightarrow\left[{}\begin{matrix}x=1+\sqrt{377}\\x=1-\sqrt{377}\end{matrix}\right.\)

`x^2 = 1000`

\(x^2=\left(\pm10\right)^2\)

`=> x = {-10;10}`

X=500

???

`a)x^2-9b^2+y^2+2xy`

`=(x^2+2xy+y^2)-9b^2`

`=(x+y)^2-(3b)^2`

`=(x+y-3b)(x+y+3b)`

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`b)(3x+2)^2-(5x+4)^2`

`=(3x+2-5x-4)(3x+2+5x+4)`

`=(-2x-2)(8x+6)`

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`c)x^2-1+4y^2-4xy`

`=(x^2-4xy+4y^2)-1`

`=(x-2y)^2-1^2`

`=(x-2y-1)(x-2y+1)`

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`d)(x+2)^2-x^2+2x-1`

`=(x+2)^2-(x^2-2x+1)`

`=(x+2)^2-(x-1)^2`

`=(x+2-x+1)(x+2+x-1)`

`=3(2x+1)`