2^(2x)=1/81

Solution for 2^(2x)=1/81 equation:

2^(2x)=1/81

We move all terms to the left:

2^(2x)-(1/81)=0

We add all the numbers together, and all the variables

2^2x-(+1/81)=0

We get rid of parentheses

2^2x-1/81=0

We multiply all the terms by the denominator


2^2x*81-1=0

Wy multiply elements

162x^2-1=0

a = 162; b = 0; c = -1;
Δ = b2-4ac
Δ = 02-4·162·(-1)
Δ = 648
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{648}=\sqrt{324*2}=\sqrt{324}*\sqrt{2}=18\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-18\sqrt{2}}{2*162}=\frac{0-18\sqrt{2}}{324}
=-\frac{18\sqrt{2}}{324}
=-\frac{\sqrt{2}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+18\sqrt{2}}{2*162}=\frac{0+18\sqrt{2}}{324}
=\frac{18\sqrt{2}}{324}
=\frac{\sqrt{2}}{18}
$