x raised by the power of y equals infinity.

Question

How many zeros are there in this equation?

10000^999

In my calculator it says infinity but that doesn't seem right.

Answer 1

The answer is: $3996$.

$10000^1 - 4$ zeroes $(4 \cdot 1)$

$10000^2 - 8$ zeroes $(4 \cdot 2)$

...

$10000^{999} - 3996$ zeroes $(4 \cdot 999)$

Answer 2

Everytime we mulltiply a number with $10$ we get:

$$n \cdot 10 = \overline{n0}$$

This would hold for any power of $10$. So for $1000=10^4$, this would be:

$$n \cdot 10000 = \overline{n0000}$$

If we multiply again we have:

$$\overline{n0000} \cdot 10000 = \overline{n00000000}$$

To sum up everytime we multiply we add $k$ zeroes at the end of the number, where $k$ is the exponent in the prime power of $10$.

So in our specific case, take $n=1$ and multiply by $10^4$ that would add $4$ zeroes at the end of the number, repeat that $999$ times and you'll get that $10000^{999}$ will have $4\cdot 999 = 3996$ zeroes.

Note that this property is a consequence of our decimal numbercal system. If we were working in a numberical system with base $k$ then the same property would hold for every power of $k$.