Oh, dude, prime factorization, really? Alright, so 22 can be broken down into 2 x 11. To write it with exponents, you'd have 2^1 x 11^1. So, there you go, prime factorization of 22 with a sprinkle of exponents.
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22 is 11 x 2 and both 11 and 2 are prime numbers. So we write the prime factorization as 2 x 11. Now the question asks for exponents, but it appears none are there. In fact, 11 can be also written as 111 and the same with 2. So you can write 22 = 111 x 21 .
Most people don't write it this way, but it does tell you that there is one 11 and one 2 in the prime factorization.
You don't find the prime factorization by using exponents, they're just a shorthand for notation once you find it. The prime factorization of 900 is 22 x 32 x 52
212 = 22 x 531
22 x 17 = 68
22 x 7 = 28
22 x 34 x 52 =8100