Write down the prime factorisation of 48 in index form, and show your working in a factor tree

Write down the prime factorisation of 48 in index form, and show your working in a factor tree.

The correct answer and explanation is:

✅ Prime Factorisation of 48 in Index Form

The prime factorisation of 48 is:
48 = 2⁴ × 3

🌲 Factor Tree of 48

Here’s how a factor tree would break down 48 into its prime factors:

        48
       /  \
      6    8
     / \   / \
    2  3  2  4
            / \
           2  2

Now gather all the prime numbers from the ends of the branches:
2, 3, 2, 2, 2
Group the 2s together:
2 × 2 × 2 × 2 × 3 = 2⁴ × 3

🧠 300-Word Explanation

Prime factorisation is the process of expressing a whole number as the product of its prime numbers—numbers that can only be divided by 1 and themselves (like 2, 3, 5, 7, 11, etc.). In the case of 48, the goal is to break it down until every factor is a prime number.

To do this, we can use a factor tree, which is a visual method of finding all the prime factors of a number. We start by splitting 48 into two smaller factors. For example, 48 can be broken down into 6 and 8. These are not prime numbers, so we break them down further:

• 6 = 2 × 3 (both prime)
• 8 = 2 × 4 (4 is not prime, so split again: 4 = 2 × 2)

At the end, all branches lead to prime numbers: 2, 2, 2, 2, and 3. We count how many times each prime number appears:

• 2 appears four times
• 3 appears once

So, in index form (also called exponential form), the prime factorisation of 48 is written as:
2⁴ × 3

This method ensures the number is broken down into its simplest building blocks. Factor trees can vary in structure (e.g., starting with 4 × 12 or 2 × 24), but they will always lead to the same prime factors, proving that prime factorisation is unique for each number.