4^n*8^n+1=16

Solution for 4^n*8^n+1=16 equation:

4^n*8^n+1=16

We move all terms to the left:

4^n*8^n+1-(16)=0

We add all the numbers together, and all the variables

4^n*8^n-15=0

Wy multiply elements

32n^2-15=0

a = 32; b = 0; c = -15;
Δ = b2-4ac
Δ = 02-4·32·(-15)
Δ = 1920
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:

$\sqrt{\Delta}=\sqrt{1920}=\sqrt{64*30}=\sqrt{64}*\sqrt{30}=8\sqrt{30}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-8\sqrt{30}}{2*32}=\frac{0-8\sqrt{30}}{64}
=-\frac{8\sqrt{30}}{64}
=-\frac{\sqrt{30}}{8}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+8\sqrt{30}}{2*32}=\frac{0+8\sqrt{30}}{64}
=\frac{8\sqrt{30}}{64}
=\frac{\sqrt{30}}{8}
$