Write the prime factorization of 392 in exponent form.
Before we can write the prime factorization in exponent form, we’ll need to find the prime factorization of 392. We notice that 392 is even. And that means we know it’s going to have a factor of two. It will be divisible by two. To divide 392 by two, we say that two goes into three one time with the remainder of one. And then we say two goes into 19 nine times. Nine times two is 18. 19 minus 18 will have a remainder of one. And finally, we’ll ask how many times does two go into 12, which is six times. And that means two times 196 equals 392.
But again, we have an even-numbered factor, which means we know we can take out another prime factor of two. If we divide 196 by two, we’ll use the same process. Two goes into 19 nine times with the remainder of one. Two goes into 16 eight times with no remainder. So two times 98 equals 196.
And here again we have another even number that we can divide by two. Two goes into nine four times with the remainder of one. And two goes into 18 nine times with the remainder of zero. And so we can say that two times 49 equals 98.
49 is not an even number, and therefore two is no longer a factor. However, we do recognize 49 as a square number. 49 is equal to seven times seven. Two is prime, and seven is prime, which means we found the prime factorization of 392. It’s two times two times two times seven times seven.
And now we want to take this repeated multiplication and turn them into exponents. In the prime factorization, we have two times two times two, which can be represented in exponent form as two cubed. And in the same way, seven times seven can be represented as seven squared. The exponential form of the prime factorization of 392 is then two cubed times seven squared.