Powerful Zeros
How many zeros are there at the end of $3^4 \times 4^5 \times 5^6$?
Age
14 to 16
Challenge level
Problem
Imagine the number $3^4\times 4^5\times 5^6$ is written out in full. How many zeros are there at the end of the number?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 6 zeros
Multiplying by $10$, or multiples/powers of $10$, makes zeros, so find powers of $10$
$3^4\times 4^5\times 5^6$
$=3^4\times 2^{10}\times 5^6$
$=3^4\times 2^4\times 2^6 \times 5^6$
$=\underbrace{6^4}_{\text{no zeros}}\times\underbrace{10^6}_{\text{six zeros}}$