Write 48 as a product of its prime factors. Write your answer in index form.
To write a number as the product of its prime factors, we must first find what is prime factors are. The prime factors of 48 are the factors of 48 which have exactly two factors of their own.
To do this, we can use a factor tree. We begin with 48 and we think of any pair of factors, whether prime or not, which multiply to give 48.
Let’s try four and 12. Now, neither of these are prime numbers as they have more than two factors. So we need to continue our factor tree. Four is the product of two and two; it’s equal to two times two.
Two is a prime number. So we circle each two. And this branch of the factor tree stops here. Two cannot be broken down any further as it’s already prime.
A common mistake would be to think that we can continue the factor tree by writing two as the product of one and two. Now, this is true. But one is not a prime number as it has only one factor, one. So our factor tree stops when we arrive at the prime number two.
On the other branch of the factor tree, 12 can be written as the product of two and six. As we’ve already said, two is prime. So we circle it. But six isn’t. So we continue this branch of the factor tree.
Six is equal to two multiplied by three, both of which are prime numbers. So we circle them both. And our factor tree is complete.
To write 48 as the product of its prime factors, we need to collect up all of the prime factors on our factor tree and multiply them together. We have four twos and one three. So we have two multiplied by two multiplied by two multiplied by two multiplied by three.
It’s usual to write the list of factors in ascending order. And in fact, our factors are already in ascending order when we look at the factor tree. We’re nearly done. But notice that the question asks to write our answer in index form. That means involving powers.
Two times two times two times two can be written as two to the power of four. So we can write our answer as two to the power of four multiplied by three. And this is 48 as a product of its prime factors in index form.
Now, it is worth just mentioning that we would have got the same answer regardless of the pair of factors that we started with. So suppose instead of four and 12, we started with two and 24.
Two is prime. But 24 isn’t and can be written as the product of two and 12. 12 can be written as the product of two and six. And six can be written as the product of two and three.
Once again, we have four factors of two on our factor tree and one factor of three. So our answer would still be two to the power of four multiplied by three.
It doesn’t matter what pair of factors that you start with or indeed how you break down any of the nonprime numbers in your factor tree. You still get the same answer at the end.