8^2n=1/64

Solution for 8^2n=1/64 equation:

8^2n=1/64

We move all terms to the left:

8^2n-(1/64)=0

We add all the numbers together, and all the variables

8^2n-(+1/64)=0

We get rid of parentheses

8^2n-1/64=0

We multiply all the terms by the denominator


8^2n*64-1=0

Wy multiply elements

512n^2-1=0

a = 512; b = 0; c = -1;
Δ = b2-4ac
Δ = 02-4·512·(-1)
Δ = 2048
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:

$\sqrt{\Delta}=\sqrt{2048}=\sqrt{1024*2}=\sqrt{1024}*\sqrt{2}=32\sqrt{2}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-32\sqrt{2}}{2*512}=\frac{0-32\sqrt{2}}{1024}
=-\frac{32\sqrt{2}}{1024}
=-\frac{\sqrt{2}}{32}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+32\sqrt{2}}{2*512}=\frac{0+32\sqrt{2}}{1024}
=\frac{32\sqrt{2}}{1024}
=\frac{\sqrt{2}}{32}
$