Index Notation
Fold a sheet of paper in half again and again and the layers pile up frighteningly fast:
2, 4, 8, 16, 32, \dots A computer's memory, a colony of bacteria,
the grains of rice on a chessboard — all grow by multiplying the same number over and
over. Index notation is the neat shorthand for writing exactly that.
When we
multiply
the same number by itself again and again, writing it out in full gets long fast.
Index notation is the shorthand:
2 \times 2 \times 2 = 2^3
We read 2^3 as “two to the power of three”. The big number
at the bottom is the base — the number being multiplied. The little number
raised up high is the index (also called the exponent or
power) — it counts how many copies of the base are multiplied
together.
So the index is a counting number, not a multiplying number.
5^3 means three fives multiplied:
5^3 = 5 \times 5 \times 5 = 125
Notice 5^3 is not
5 \times 3 = 15. The little
3 never gets multiplied in — it just tells you to write
down three fives.
Reading and saying powers
Out loud we say “base to the power of index”, but two powers are so common they
have their own short names:
-
n^2 is read “n squared” — because it gives the
area of a square with side n.
-
n^3 is read “n cubed” — because it gives the
volume of a cube with side n.
So all of these mean exactly the same thing:
4^2 = 4 \times 4 = 16 \qquad\text{“four squared is sixteen.”}
Make a square that is 4 stars wide and
4 stars tall. How many stars fit inside? It is
4 rows of 4:
Sixteen stars — and the shape really is a square. That is why
4 \times 4 is called “four squared”, written
4^2 = 16.
See it: a number squared is a square
Here is a square made of dots. It is always the same number of dots across as
it is down — that is what squared means. Count a row, then there are
that many rows. Press Refresh for a new square.
Worked examples
Take each one slowly: write out the copies, then multiply.
-
2^4 = 2 \times 2 \times 2 \times 2 = 16 — four copies of
2.
-
3^3 = 3 \times 3 \times 3 = 27 — three cubed.
-
10^2 = 10 \times 10 = 100 — ten squared is one hundred.
A small index can grow into a big number surprisingly fast:
2^{10} = 1024 — that is just ten twos multiplied together.
A cube has the same length, width and height. To find how many little boxes fill a
cube that is 2 on every side, build it in layers. The bottom
layer is a 2 \times 2 square of oranges, and the top layer is
another one stacked on top:
bottom layer top layer
Two layers of four oranges make eight in all:
2^3 = 2 \times 2 \times 2 = 8. That is why
2^3 is called “two cubed” — it fills a little cube.
Powers of ten and place value
Powers of 10 are special friends of
place value.
Watch what happens:
10^1 = 10 \qquad 10^2 = 100 \qquad 10^3 = 1000 \qquad 10^4 = 10000
The index tells you exactly how many zeros follow the
1! 10^3 has three zeros, which is a
thousand. Each step up the powers of ten is a brand-new place-value column: ones, tens,
hundreds, thousands. That is why our whole number system is built from powers of ten.
The two index traps that catch everyone:
-
5^3 means 5 \times 5 \times 5 = 125,
not 5 \times 3 = 15. The index is a
count of copies — never a number you multiply by.
-
3^2 = 9 (three squared), not
3 \times 2 = 6. If in doubt, write out the copies in full
first, then multiply.
Khan Academy introduces exponents here: