Expanded notation with 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.

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.

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.

The exponent and the zeros

The powers of ten hide a tidy pattern that makes them easy to read. Ten to the power 0 is 1, ten to the 1 is 10, ten squared is 100, ten cubed is 1000, and the run carries on. Each step up in the exponent adds one more zero, so the exponent is simply a count of the zeros after the 1. That is no accident: raising the power by one multiplies by another ten, which lengthens the number by a single zero every time. Spotting this turns reading big powers of ten into a glance rather than a calculation.

What multiplying by ten really does

Because each column is ten times the one to its right, multiplying a whole number by ten slides every digit one column to the left and leaves a zero in the ones place. The 47 that sat in the tens and ones becomes 470 in the hundreds and tens, then 4700, then 4700 and on, the same two digits travelling leftward through the powers of ten. Dividing by ten reverses the journey, moving each digit one column back to the right. This single idea, that times ten is a shift of place, sits underneath much of the arithmetic of large numbers.

Putting the number back together

Expanded notation is a two-way street. Taking a number apart writes it as a sum of place values, and reading that sum in reverse rebuilds the number. Six thousands, two tens and four ones assemble into 6024, but only if a zero steps in to hold the empty hundreds place. Leaving the zero out would let the other digits slide together and quietly change 6024 into 624. Being able to travel in both directions, expanding a number and reassembling it, is what makes place value a tool you can rely on rather than a rule you simply recite.