1/6^2n=36

Solution for 1/6^2n=36 equation:

1/6^2n=36

We move all terms to the left:

1/6^2n-(36)=0


Domain of the equation: 6^2n!=0

n!=0/1

n!=0

n∈R


We multiply all the terms by the denominator


-36*6^2n+1=0

Wy multiply elements

-216n^2+1=0

a = -216; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-216)·1
Δ = 864
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:

$\sqrt{\Delta}=\sqrt{864}=\sqrt{144*6}=\sqrt{144}*\sqrt{6}=12\sqrt{6}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-12\sqrt{6}}{2*-216}=\frac{0-12\sqrt{6}}{-432}
=-\frac{12\sqrt{6}}{-432}
=-\frac{\sqrt{6}}{-36}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+12\sqrt{6}}{2*-216}=\frac{0+12\sqrt{6}}{-432}
=\frac{12\sqrt{6}}{-432}
=\frac{\sqrt{6}}{-36}
$