Question

urgent
Rezolvati:
1.
$(\frac{1}{5} ) ^{x} \times 3x = \sqrt{ \frac{27}{125} }$
2.
$4 \times ( \frac{1}{16} ) ^{x} + 15 \times ( \frac{ 1}{4} )^{x} - 4 = 0$
3.
$3 ^{x} = - x - \frac{ 2}{3}$
​

Answer 1

1.

$\it \Big(\dfrac{1}{5}\Big)^x\cdot3^x=\sqrt[3]{\dfrac{27}{125}} \Rightarrow \Big(\dfrac{3}{5}\Big)^x=\dfrac{3}{5} \Rightarrow x=1$

2.

$\it 4\cdot\Big(\dfrac{1}{16}\Big)^x+15\cdot\Big(\dfrac{1}{4}\Big)^x-4=0 \Rightarrow 4\cdot\Big(\dfrac{1}{4}\Big)^{2x}+15\cdot\Big(\dfrac{1}{4}\Big)^x-4=0\\ \\ \\ Not\breve am\ \Big(\dfrac{1}{4}\Big)^x=t,\ t>0,\ \ iar\ \ ecua\c{\it t}ia\ devine:\\ \\ 4t^2+15t-4=0 \Rightarrow 4t^2+16t-t-4=0 \Rightarrow 4t(t+4)-(t+4)=0 \Rightarrow \\ \\ \Rightarrow (t+4)(4t-1)=0 \Rightarrow t_1=-4<0\ (nu\ convine),\ \ t_2=\dfrac{1}{4}$

Revenim asupra notației și obținem:

$\it t=\dfrac{1}{4} \Rightarrow \Big(\dfrac{1}{4}\Big)^x=\dfrac{1}{4} \Rightarrow x=1$