3^(2y)=(1)/(81)

Solution for 3^(2y)=(1)/(81) equation:

3^(2y)=(1)/(81)

We move all terms to the left:

3^(2y)-((1)/(81))=0

We add all the numbers together, and all the variables

3^2y-(+1/81)=0

We get rid of parentheses

3^2y-1/81=0

We multiply all the terms by the denominator


3^2y*81-1=0

Wy multiply elements

243y^2-1=0

a = 243; b = 0; c = -1;
Δ = b2-4ac
Δ = 02-4·243·(-1)
Δ = 972
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{972}=\sqrt{324*3}=\sqrt{324}*\sqrt{3}=18\sqrt{3}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-18\sqrt{3}}{2*243}=\frac{0-18\sqrt{3}}{486}
=-\frac{18\sqrt{3}}{486}
=-\frac{\sqrt{3}}{27}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+18\sqrt{3}}{2*243}=\frac{0+18\sqrt{3}}{486}
=\frac{18\sqrt{3}}{486}
=\frac{\sqrt{3}}{27}
$