1/9^a+1=81^a

Solution for 1/9^a+1=81^a equation:

1/9^a+1=81^a

We move all terms to the left:

1/9^a+1-(81^a)=0


Domain of the equation: 9^a!=0

a!=0/1

a!=0

a∈R


We multiply all the terms by the denominator


-81^a*9^a+1*9^a+1=0

Wy multiply elements

-729a^2+9a+1=0

a = -729; b = 9; c = +1;
Δ = b2-4ac
Δ = 92-4·(-729)·1
Δ = 2997
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2997}=\sqrt{81*37}=\sqrt{81}*\sqrt{37}=9\sqrt{37}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(9)-9\sqrt{37}}{2*-729}=\frac{-9-9\sqrt{37}}{-1458}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(9)+9\sqrt{37}}{2*-729}=\frac{-9+9\sqrt{37}}{-1458}
$