3(2/3)n=2/9

Solution for 3(2/3)n=2/9 equation:

3(2/3)n=2/9

We move all terms to the left:

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


Domain of the equation: 3)n!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

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

We multiply parentheses

6n^2-(+2/9)=0

We get rid of parentheses

6n^2-2/9=0

We multiply all the terms by the denominator


6n^2*9-2=0

Wy multiply elements

54n^2-2=0

a = 54; b = 0; c = -2;
Δ = b2-4ac
Δ = 02-4·54·(-2)
Δ = 432
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:

$\sqrt{\Delta}=\sqrt{432}=\sqrt{144*3}=\sqrt{144}*\sqrt{3}=12\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-12\sqrt{3}}{2*54}=\frac{0-12\sqrt{3}}{108}
=-\frac{12\sqrt{3}}{108}
=-\frac{\sqrt{3}}{9}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+12\sqrt{3}}{2*54}=\frac{0+12\sqrt{3}}{108}
=\frac{12\sqrt{3}}{108}
=\frac{\sqrt{3}}{9}
$