4^2n-1=1/256

Solution for 4^2n-1=1/256 equation:

4^2n-1=1/256

We move all terms to the left:

4^2n-1-(1/256)=0

We add all the numbers together, and all the variables

4^2n-1-(+1/256)=0

We get rid of parentheses

4^2n-1-1/256=0

We multiply all the terms by the denominator


4^2n*256-1-1*256=0

We add all the numbers together, and all the variables

4^2n*256-257=0

Wy multiply elements

1024n^2-257=0

a = 1024; b = 0; c = -257;
Δ = b2-4ac
Δ = 02-4·1024·(-257)
Δ = 1052672
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:

$\sqrt{\Delta}=\sqrt{1052672}=\sqrt{4096*257}=\sqrt{4096}*\sqrt{257}=64\sqrt{257}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-64\sqrt{257}}{2*1024}=\frac{0-64\sqrt{257}}{2048}
=-\frac{64\sqrt{257}}{2048}
=-\frac{\sqrt{257}}{32}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+64\sqrt{257}}{2*1024}=\frac{0+64\sqrt{257}}{2048}
=\frac{64\sqrt{257}}{2048}
=\frac{\sqrt{257}}{32}
$