You are a knowledgeable mathematics tutor with a passion for making complex concepts easy to understand. You possess the ability to break down mathematical statements into simpler components, providing clear explanations and examples that cater to various learning styles.

Your task is to explain a specific mathematical statement. Here is the statement I need help with:

explain "So I have this problem here: find (21^2 mod 15)^3 mod 22. Hmm, okay, let's break this down step by step. I need to compute this expression, and it looks like it involves modular arithmetic, which can sometimes be tricky, but if I take it one step at a time, I should be able to manage.

First, I need to compute 21 squared, then take that result modulo 15, and then cube that result, and finally take that modulo 22. So, let's start with the first part: 21 squared.

What's 21 squared? 21 times 21. Well, 20 times 20 is 400, and then 20 times 1 is 20, so 20 times 21 is 420. Then, 1 times 21 is 21. So, 420 plus 21 is 441. So, 21 squared is 441.

Now, I need to take 441 modulo 15. Modulo operation gives the remainder when one number is divided by another. So, I need to divide 441 by 15 and find the remainder.

Let's see, how many times does 15 go into 441? Well, 15 times 20 is 300, and 15 times 29 is 435, because 15 times 30 would be 450, which is too big. So, 15 times 29 is 435. Subtracting 435 from 441 gives a remainder of 6. So, 441 modulo 15 is 6.

Now, I need to take that result, which is 6, and cube it. So, 6 cubed is 6 times 6 times 6. 6 times 6 is 36, and 36 times 6 is 216. So, 6 cubed is 216.

Finally, I need to take 216 modulo 22. Again, this is the remainder when 216 is divided by 22. Let's see, 22 times 9 is 198, and 22 times 10 is 220, which is too big. So, 22 times 9 is 198. Subtracting 198 from 216 gives 18. So, 216 modulo 22 is 18.

Therefore, the final answer should be 18.

Wait a minute, let me double-check that to make sure I didn't make any mistakes. Starting from the beginning:

21 squared is indeed 441.

441 divided by 15 is 29 with a remainder of 6, since 15 times 29 is 435, and 441 minus 435 is 6.

Then, 6 cubed is 216.

216 divided by 22 is 9 with a remainder of 18, since 22 times 9 is 198, and 216 minus 198 is 18.

So, yes, it seems correct. I think the answer is 18.

Final Answer

[ \boxed{18} ]"