(1/5)n-1/3=8/3

Solution for (1/5)n-1/3=8/3 equation:

(1/5)n-1/3=8/3

We move all terms to the left:

(1/5)n-1/3-(8/3)=0


Domain of the equation: 5)n!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

(+1/5)n-1/3-(+8/3)=0

We multiply parentheses

n^2-1/3-(+8/3)=0

We get rid of parentheses

n^2-1/3-8/3=0

We multiply all the terms by the denominator


n^2*3-1-8=0

We add all the numbers together, and all the variables

n^2*3-9=0

Wy multiply elements

3n^2-9=0

a = 3; b = 0; c = -9;
Δ = b2-4ac
Δ = 02-4·3·(-9)
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-6\sqrt{3}}{2*3}=\frac{0-6\sqrt{3}}{6}
=-\frac{6\sqrt{3}}{6}
=-\sqrt{3}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+6\sqrt{3}}{2*3}=\frac{0+6\sqrt{3}}{6}
=\frac{6\sqrt{3}}{6}
=\sqrt{3}
$