9^2n=1/3

Solution for 9^2n=1/3 equation:

9^2n=1/3

We move all terms to the left:

9^2n-(1/3)=0

We add all the numbers together, and all the variables

9^2n-(+1/3)=0

We get rid of parentheses

9^2n-1/3=0

We multiply all the terms by the denominator


9^2n*3-1=0

Wy multiply elements

27n^2-1=0

a = 27; b = 0; c = -1;
Δ = b2-4ac
Δ = 02-4·27·(-1)
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-6\sqrt{3}}{2*27}=\frac{0-6\sqrt{3}}{54}
=-\frac{6\sqrt{3}}{54}
=-\frac{\sqrt{3}}{9}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+6\sqrt{3}}{2*27}=\frac{0+6\sqrt{3}}{54}
=\frac{6\sqrt{3}}{54}
=\frac{\sqrt{3}}{9}
$