Question

Помогите пожалуйста,заранее спасибо.
Решить уравнения 1)$7*3 ^{x+1} -5 ^{x+2} = 3^{x+4} -5 ^{x+3}$
2)$( sqrt{2- sqrt{3} } ) ^{x} +( sqrt{2+ sqrt{3} } ) ^{x}=4$
3)$3*4 ^{x} -5*6 ^{x} +2*9 ^{x} =0$
4)$125*25 ^{x} -70*10 ^{x} +8*4 ^{x}$

Answer 1

1) $5^{x+3} -5^{x+2}=3^{x+4}-7*3^{x+1}$
$5^{x+1}(25-5 )=3^{x+1}(27-7)$
$( frac{5}{3})^{x+1} = frac{20}{20} =1$
$x+1=0$
$x=-1$

2)$( sqrt{2+ sqrt{3} } ) ^{x} =t, t textgreater 0$
$( sqrt{2- sqrt{3} } ) ^{x}= frac{1}{t}$
$frac{1}{t} +t=4$
$t^{2} -4t+1=0$
$t_{1} = 2+sqrt{4-1}=2+ sqrt{3}$
$(2+ sqrt{3}) ^{ frac{x}{2} } = (2+ sqrt{3})^1 }$
$frac{x}{2} =1$
$x=2$
$t_{2} =2- sqrt{3}$
$(2- sqrt{3}) ^{ frac{x}{2} } = (2- sqrt{3})^{-1}$
$frac{x}{2} =-1$
$x=-2$
$x_{1}=-2 , x_{2} =2$

3)$3* 2^{2x}-5 (2*3)^{x} +2* 3^{2x}=0 |/ 3^{2x}$
$3* ( frac{2}{3} )^{2x} -5*( frac{2}{3} )^{x}+2=0$
$( frac{2}{3} )^{x}=t, t textgreater 0$
$( frac{2}{3} )^{2x}= t^{2}$
$3t-5t+2=0$
$t_{1} = frac{2}{3}$
$( frac{2}{3} )^{x} = frac{2}{3}$
$x=1$
$t_{2} =1$
$( frac{2}{3} )^{x} =(frac{2}{3}) ^{0}$
$x=0$
$x_{1}=0 , x_{2} =1$

4)$125* 5^{2x}-70*(2*5)^{x} +8* 2^{2x}=0 |/ 5^{2x}$
$125-70*( frac{2}{5} )^{x}+8* ( frac{2}{5} )^{2x} =0$
$( frac{2}{5} )^{x}=t, t textgreater 0$
$( frac{2}{5} )^{2x}= t^{2}$
$8 t^{2}-70t+125=0$
$t_{1} = frac{5}{2}$
$( frac{2}{5} )^{x} = (frac{2}{5}) ^{-1}$
$x=-1$
$t_{2} = frac{25}{4}$
$( frac{2}{5} )^{x} = (frac{2}{5}) ^{-2}$
$x=-2$
$x_{1}=-2 , x_{2} =-1$