i understood a,b and c but d made not alot of sense. how do i go backwards? thanks
It's better to think about the exponents as fractions.
If $\displaystyle 2^2=4$
Then $\displaystyle 4^\frac{1}{2}=2$
If $\displaystyle 9^4=6561$
Then $\displaystyle 6561^\frac{1}{4}=9$
So it's easiest to work backwards.
$\displaystyle 3^4=81$
So $\displaystyle 81^?=3$
$\displaystyle 125^x=25^{x+1}$
I would do that by writing both as 5^{something}
$\displaystyle 125=5^3$
$\displaystyle 25=5^2$
So $\displaystyle (5^3)^x=(5^2)^{x+1}$
$\displaystyle 5^{3x}=5^{2x+2}$
So:
$\displaystyle 3x=2x+2$
$\displaystyle x=2$
Part b) is harder. Try to write it out as powers of 3.
For part c) Try to write it out as powers of 2
Don't confuse the OP, integral (even your signature seems to be an extension of your answer xD)
For part c), you could note that $\displaystyle 4 = 2 \times 2$, and try to find yourself with just two's on the right side. Then note that $\displaystyle 8 = 2^3$. The rest should follow.