I have a question that is asking me "How many factors does each of the following numbers have?
12^4=
12^4 = 12*12*12*12
12^4 = (2^2*3)^4 = 2^8 3^4
so, there are 9*5=45 factors
To determine how many factors 12^4 has, we need to understand the factors of this number.
The expression 12^4 means 12 raised to the power of 4, which is equivalent to 12 × 12 × 12 × 12.
To find the factors, we can factorize 12. The prime factorization of 12 is 2^2 × 3^1, which means 12 can be expressed as 2^2 × 3^1.
Now, let's find the factors of 12^4. We raise each prime factor to the 4th power:
(2^2 × 3^1) ^ 4 = 2^8 × 3^4.
So, the prime factorization of 12^4 is 2^8 × 3^4.
To find the factors, we need to consider all possible combinations of the prime factors.
For the powers of 2, we have 0, 1, 2, ..., 8. That gives us a total of 9 possibilities.
For the powers of 3, we have 0, 1, 2, ..., 4. That gives us a total of 5 possibilities.
To find the total number of factors, we multiply these possibilities together: 9 × 5 = 45.
Therefore, 12^4 has 45 factors.