1/3n=22n

Solution for 1/3n=22n equation:

1/3n=22n

We move all terms to the left:

1/3n-(22n)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

-22n+1/3n=0

We multiply all the terms by the denominator


-22n*3n+1=0

Wy multiply elements

-66n^2+1=0

a = -66; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-66)·1
Δ = 264
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{264}=\sqrt{4*66}=\sqrt{4}*\sqrt{66}=2\sqrt{66}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{66}}{2*-66}=\frac{0-2\sqrt{66}}{-132}
=-\frac{2\sqrt{66}}{-132}
=-\frac{\sqrt{66}}{-66}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{66}}{2*-66}=\frac{0+2\sqrt{66}}{-132}
=\frac{2\sqrt{66}}{-132}
=\frac{\sqrt{66}}{-66}
$