81^2k=1/243

Solution for 81^2k=1/243 equation:

81^2k=1/243

We move all terms to the left:

81^2k-(1/243)=0

We add all the numbers together, and all the variables

81^2k-(+1/243)=0

We get rid of parentheses

81^2k-1/243=0

We multiply all the terms by the denominator


81^2k*243-1=0

Wy multiply elements

19683k^2-1=0

a = 19683; b = 0; c = -1;
Δ = b2-4ac
Δ = 02-4·19683·(-1)
Δ = 78732
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{78732}=\sqrt{26244*3}=\sqrt{26244}*\sqrt{3}=162\sqrt{3}$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-162\sqrt{3}}{2*19683}=\frac{0-162\sqrt{3}}{39366}
=-\frac{162\sqrt{3}}{39366}
=-\frac{\sqrt{3}}{243}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+162\sqrt{3}}{2*19683}=\frac{0+162\sqrt{3}}{39366}
=\frac{162\sqrt{3}}{39366}
=\frac{\sqrt{3}}{243}
$