If an astronaut can throw a certain wrench 10.0 m vertically upward on earth, how high could he throw it on our moon if he gives it the same starting speed in both places?

I think you forgot to include the acceleration due to gravity of astronauts. I assume that it is = 0.170 g. To get the answer we have to use the formula s = v0t – (1/2) At². Where s is the altitude, A is the acceleration of gravity, t is the time after throwing.

v = v0 –At

v = 0 at max altitude so v0 – At = 0

t = v0/A at max altitude

Using the formula above for the altitude:

s = v0t – (1/2) At²

s = v0(v0/A) – (1/2) A (v0/A)²

s = v0²/A – (1/2) v0²/A

s = (1/2) v0²/A

The earth: E = (1/2) v0²/g

The moon: M = (1/2)v0²(0.17g)

So, take the ratio of M/E = g/0.17g = 1/0.17 = 588

M = 5.88 E

He can throw the wrench 5.88 times higher on the moon

it is called kinetic energy