150=a*1.66666667^8

Solution for 150=a*1.66666667^8 equation:

150=a*1.66666667^8

We move all terms to the left:

150-(a*1.66666667^8)=0

We get rid of parentheses

-a*1.66666667^8+150=0

Wy multiply elements

-1a^2+150=0

a = -1; b = 0; c = +150;
Δ = b2-4ac
Δ = 02-4·(-1)·150
Δ = 600
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{600}=\sqrt{100*6}=\sqrt{100}*\sqrt{6}=10\sqrt{6}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-10\sqrt{6}}{2*-1}=\frac{0-10\sqrt{6}}{-2}
=-\frac{10\sqrt{6}}{-2}
=-\frac{5\sqrt{6}}{-1}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+10\sqrt{6}}{2*-1}=\frac{0+10\sqrt{6}}{-2}
=\frac{10\sqrt{6}}{-2}
=\frac{5\sqrt{6}}{-1}
$