AC9M7N03 · Year 7 · Number

Expanded notation with powers of 10

ACARA v9 CONTENT DESCRIPTION “represent natural numbers in expanded notation using place value and powers of 10”

Our entire number system is built on the number ten. The position of a digit tells you how much it is worth, and each position is exactly ten times the one to its right. This is so familiar that we rarely think about it, but writing numbers out in full, using powers of ten, reveals the clever structure hiding inside every number you read.

This year you learn to represent whole numbers in expanded notation, showing the value of each digit, and to express those place values as powers of ten. It connects the everyday skill of reading large numbers to the exponent notation you have just met, and it lays the groundwork for understanding very large and very small numbers later on.

Place value as powers of ten

In a number like 3705, each digit lives in a column with its own value. The 5 is in the ones column, the 0 in the tens, the 7 in the hundreds and the 3 in the thousands. What makes the system elegant is that each of these columns is a power of ten. The ones column is 10 to the power 0, which equals 1. The tens column is 10 to the power 1, the hundreds is 10 squared, and the thousands is 10 cubed. Every step to the left raises the power by one and multiplies the value by another ten.

Each place is a power of ten
Moving one column left multiplies the place value by ten.
Each column in a number has a value that is a power of ten. The ones column is 10 to the power 0, which is 1, and every step to the left multiplies by another ten, giving tens, hundreds, thousands, and so on.

Seeing the columns as powers of ten explains a pattern you may have noticed: the number of zeros in a place value matches its power. Ten cubed is 1000, with three zeros. This is not a coincidence but a direct consequence of multiplying by ten each time. Understanding this makes reading and comparing large numbers far less about memorising and far more about recognising a simple, repeating structure.

Writing a number in expanded form

Expanded notation takes a number apart and writes it as the sum of what each digit contributes. For 3705 this is 3 times 1000, plus 7 times 100, plus 0 times 10, plus 5 times 1. The zero in the tens place contributes nothing, which is exactly why we can leave it out of the final sum, though the zero still matters for holding the other digits in their correct columns.

Expanding a number
Expanded notation splits a number into the value contributed by each digit.
Expanded notation rewrites a number as the sum of what each digit is worth. The 3 in 3705 is worth 3 thousands, the 7 is worth 7 hundreds, and the 5 is worth 5 ones. Written with powers of ten, this becomes 3 times 10 cubed plus 7 times 10 squared plus 5.

The same expansion can be written with powers of ten in place of the plain values, giving 3 times 10 cubed, plus 7 times 10 squared, plus 5. Reading the expansion in reverse lets you assemble a number from its parts: 6 thousands, 2 tens and 4 ones combine to make 6024, with a zero quietly holding the empty hundreds place. Moving fluently in both directions, taking a number apart and putting it back together, is the heart of place value and a skill you will lean on whenever you work with the size and structure of numbers.

Teaching tip: place value columns drawn on paper, with the powers of ten written underneath each one, turn an abstract idea into something a student can point at. Ask them to place a few numbers into the columns and then read off the expanded form, so the link between position and value becomes a habit rather than a rule to recall.

The zero is where care is needed. Stress that a zero in a column is not nothing to be ignored, but a placeholder that keeps every other digit in its proper place. Removing it would shift the digits and change the number entirely, as the difference between 6024 and 624 makes clear.

Builds on: Prime factorisation with exponents (AC9M7N02). That unit wrote repeated primes as powers; this unit uses powers of 10 to expand place value.
Quick self-check
1. What is the value of 10 to the power of 0?
2. In the number 4827, what is the place value of the 8?
3. Which is the correct expanded form of 506?
4. Written as a power of ten, what is the value of the thousands column?
5. A number is written as 6 x 1000 + 2 x 10 + 4 x 1. What is the number?