AC9M7N02 · Year 7 · Number

Prime factorisation with exponents

ACARA v9 CONTENT DESCRIPTION “represent natural numbers as products of powers of prime numbers using exponent notation”

Every whole number larger than 1 is built from prime numbers, the same way every word is built from letters. The primes are the basic building blocks of arithmetic, and any number can be written as a product of them in exactly one way. This year you learn to find those prime factors and to record them compactly using exponents.

A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11 and so on. Numbers that are not prime, like 6 or 12, are called composite, because they can be broken down into smaller factors. The goal of prime factorisation is to keep breaking a composite number down until only primes are left.

Building a factor tree

A factor tree is the simplest way to find the primes inside a number. Start with the number, split it into any two factors, then split those factors again, and keep going until every branch ends in a prime. For 60 you might split it into 6 and 10, then 6 into 2 and 3, and 10 into 2 and 5. The branches end at 2, 3, 2 and 5, the prime building blocks of 60.

A factor tree finds the primes
Split a number into factors, then split again until only primes remain.
Start with 60. It is composite, so it can be split.

It does not matter which factors you choose first. Splitting 60 into 4 and 15 instead would lead to exactly the same set of primes at the ends of the branches. This is a deep fact about numbers: the collection of primes that multiplies to a given number is always the same, no matter how you find them. That uniqueness is what makes prime factorisation so useful.

Writing repeats as powers

Once you have the primes, you often find some of them repeat. The factorisation of 60 includes the prime 2 twice. Rather than writing 2 times 2, we use an exponent and write 2 with a small raised 2, read as 2 squared. So 60 is written as 2 squared times 3 times 5. The exponent simply counts how many times a prime appears.

Exponents gather repeats
When a prime appears more than once, write it once with an exponent.
60 expands to 2 x 2 x 3 x 5. The two 2s are gathered into 2 squared, giving 2 squared x 3 x 5.

This compact form is more than just tidy. It acts as a fingerprint for the number: no other number has the prime factorisation 2 squared times 3 times 5. Writing numbers this way makes it far easier to compare them, to find common factors, and to simplify fractions, all of which rely on knowing exactly which primes a number is made from. The exponent notation keeps that information short and exact, even for very large numbers.

Teaching tip: let the student pick the first split themselves, then try a different first split for the same number and compare the leaves. Discovering that both trees end with the identical set of primes is a genuine moment of insight, and it makes the uniqueness of prime factorisation something they have seen rather than been told.

A frequent slip is stopping too early, leaving a composite number like 4 at the end of a branch. Encourage a quick check of every leaf: if a number can still be divided by anything other than 1 and itself, the tree is not finished yet.

Builds on: Square numbers and square roots (AC9M7N01). That unit wrote a number times itself using a raised exponent; this unit uses the same exponent notation to write any number as a product of its prime factors.

One number, one set of primes

The first factor pair you choose feels like it should matter, yet it never does. Split 60 as 6 times 10 and follow each branch to its primes, then split the same 60 as 4 times 15 instead: the two trees look different on the way down but arrive at the identical collection of primes, 2, 2, 3 and 5. This is the fundamental theorem of arithmetic in miniature. Every whole number above 1 has exactly one prime factorisation, which is why it can serve as a reliable fingerprint for the number.

The first split does not matter
Two different factor trees for the same number reach the very same primes.
Starting 60 as 6 x 10 still ends with the primes 2, 2, 3 and 5. The first split you pick never changes the set of primes you finish with.

Reading the factorisation backwards

A prime factorisation is meant to be reversible. If finding the primes takes a number apart, multiplying the prime powers back together must rebuild it. Expand each power first, so 2 cubed becomes 8, then carry out the multiplication: 2 cubed times 5 is 8 times 5, which is 40. Running the process in this direction is the quickest way to check a factorisation, and it confirms that the compact power form holds exactly the same value as the number it came from.

Rebuilding from the powers
Expand each power, multiply, and the original number reappears.
2 squared x 3 expands to 4 x 3, which multiplies to 12. Multiplying a prime factorisation back together always returns the number you started from.

Telling primes from composites

Before factorising, it helps to know whether a number can be broken down at all. Listing its factor pairs answers the question directly. A prime has only the single pair 1 times itself, so 7 is prime because nothing else divides it evenly. A composite number offers more than one pair, as 12 does with 1 times 12, 2 times 6 and 3 times 4, which is exactly why a factor tree has somewhere to go. Counting the factor pairs is a clean test of which numbers are the building blocks and which are built from them.

Prime or composite
List every factor pair: one pair means prime, more than one means composite.
The only way to write 7 as a product of whole numbers is 1 x 7, so 7 has exactly two factors and is prime.
Quick self-check
1. What is a prime number?
2. Which of these is the prime factorisation of 12?
3. Why do we use exponents when writing prime factorisations?
4. A factor tree is complete when
5. The prime factorisation of a number is 2 cubed x 5. What is the number?
Teaching pack: free to printReady-to-teach plans, student sheets, cut-outs and answers for this unit. Print or save as PDF.