Question

Solve: $4^{2/x}-5\left(4^{1/x}\right)+4=0$

Answer 1

We have,

$4^{2/x}-5\left(4^{1/x}\right)+4=0$

Let $4^{\frac{1}{x}}=y$ and put,

$4^{\frac{2}{x}}-5\left(4^{\frac{1}{x}}\right)+4=0$

$y^2-5y+4=0$

$y^2-(4+1)y+4=0$

$y^2-4y-y+4=0$

$y(y-4)-1(y-4)=0$

$(y-4)(y-1)=0$

If $y-4=0$ then $y=4$

Then,

$4^{\frac{1}{x}}=4$

$4^{\frac{1}{x}}=4^1$

On comparing that,

$x=1$

If $y-1=0$ then,$y=1$

Then,

$4^{\frac{1}{x}}=1$

$4^{\frac{1}{x}}=4^0$

On comparing that,

$x=0$

Hence, $(0,1)$ is the answer.