2^a+1=3^a+16^-1/a

Solution for 2^a+1=3^a+16^-1/a equation:

2^a+1=3^a+16^-1/a

We move all terms to the left:

2^a+1-(3^a+16^-1/a)=0


Domain of the equation: a)!=0

a!=0/1

a!=0

a∈R


We get rid of parentheses

2^a-3^a+1/a+1-16^=0

We multiply all the terms by the denominator


2^a*a-3^a*a+1*a-16^*a+1=0

We add all the numbers together, and all the variables

a+2^a*a-3^a*a-16^*a+1=0

Wy multiply elements

2a^2-3a^2-16a^2+a+1=0

We add all the numbers together, and all the variables

-17a^2+a+1=0

a = -17; b = 1; c = +1;
Δ = b2-4ac
Δ = 12-4·(-17)·1
Δ = 69
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-\sqrt{69}}{2*-17}=\frac{-1-\sqrt{69}}{-34}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{69}}{2*-17}=\frac{-1+\sqrt{69}}{-34}
$