Squares, Cubes and Roots
Imagine tiling a square patio: 5 tiles across and 5 tiles up.
How many tiles do you need altogether? You don't count them one by one — you do
5 \times 5 = 25. That's squaring: multiplying a
number by itself.
Now stack those tiles into a solid cube, 5 across, 5 up, 5 deep. The number of
little cubes is 5 \times 5 \times 5 = 125 — that's cubing.
And roots run the whole thing backwards: if a square has area 25, what was its
side? The square root of 25 is 5. Squares and cubes build up; roots undo them. This one
idea — powers and their inverses — is a doorway into a huge amount of maths.
Squares: numbers that draw a square
A square number is a number multiplied by itself —
n \times n = n^2 (read "n squared"). The name is no accident:
it is the exact area
of a square with side n. A row of dots for each value fills a perfect
square grid:
1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100,\ \dots
Learn these first ten by heart — they turn up everywhere, from
Pythagoras' theorem
to spotting patterns in number puzzles.
Cubes: numbers that build a cube
A cube number goes one step further — three copies multiplied together,
n \times n \times n = n^3 (read "n cubed"). It is the
volume of a cube with side n, the number of unit blocks it
takes to fill the box:
1,\ 8,\ 27,\ 64,\ 125,\ 216,\ \dots
Cubes grow fast — jumping from a side of 5 to a side of 6 leaps from 125 all the way to
216, because you're adding a whole new slab, wall and edge of cubes at once.
Roots: undoing the power
Every operation has an inverse. The square root
\sqrt{\phantom{x}} undoes squaring — it asks "what number,
squared, gives this?"
\sqrt{49} = 7 \quad\text{because}\quad 7^2 = 49
The cube root \sqrt[3]{\phantom{x}} undoes
cubing in the same way:
\sqrt[3]{27} = 3 \quad\text{because}\quad 3^3 = 27
A number whose square root is a whole number (like 49) is a perfect square;
one whose cube root is whole (like 27) is a perfect cube. Rooting a perfect
square or cube gives a clean, tidy answer — and knowing your squares and cubes lets you read the
roots straight off.
-
a square is n^2 = n \times n; a
cube is n^3 = n \times n \times n;
-
the square root \sqrt{\phantom{x}} and
cube root \sqrt[3]{\phantom{x}} are their
inverses, so \sqrt{n^2} = n and
\sqrt[3]{n^3} = n;
-
the perfect squares are
1, 4, 9, 16, 25, 36, 49, 64, 81, 100;
-
the perfect cubes are
1, 8, 27, 64, 125.
Seeing it
Step through the figure: a flat square counts its unit squares, then a cube stacks unit
cubes in three dimensions. The word "squared" is literally a picture, and "cubed" literally a
box.
Worked examples
1. A few squares and cubes. Just multiply carefully:
8^2 = 8 \times 8 = 64, \qquad 12^2 = 12 \times 12 = 144, \qquad 4^3 = 4 \times 4 \times 4 = 64.
(Notice 8^2 and 4^3 both equal 64 — a
number can be both a square and a cube!)
2. Reading roots off perfect squares. Ask "what times itself?"
\sqrt{81} = 9,\qquad \sqrt{144} = 12,\qquad \sqrt[3]{125} = 5.
3. A real patio. A square patio has side 7 m. Its
area is 7^2 = 49 m². Turn it around: if you had 49 square metres of
paving and wanted a square patio, its side would be \sqrt{49} = 7 m.
Area needs squaring; finding the side needs the square root.
What about roots that aren't tidy? Most numbers are not perfect
squares. \sqrt{2} sits somewhere between 1
(since 1^2 = 1) and 2 (since
2^2 = 4) — it's about 1.41421\ldots and the
decimals never stop and never repeat. Such a number is called irrational: it
can't be written as an exact fraction, no matter how hard you try.
This is the single most common early slip in the whole topic:
-
Squaring is not doubling. 5^2 means
5 \times 5 = 25, not 5 \times 2 = 10.
The little 2 tells you how many copies to multiply, not what to multiply by. Whenever
you see a power, say it out loud: "five, times itself".
-
Every positive number has TWO square roots. Because
5^2 = 25 and (-5)^2 = 25 (a
negative times a negative is positive), both +5 and
-5 square to 25. So "a square root of 25" could be
\pm 5. By convention the \sqrt{\phantom{x}}
symbol means just the positive one, so \sqrt{25} = 5 — but
don't forget its negative twin is lurking.
Line up the square numbers and look at the gaps between them:
1,\ 4,\ 9,\ 16,\ 25,\dots jumps by
3,\ 5,\ 7,\ 9,\dots — the odd numbers, in order!
Each new square is the last one plus the next odd number. (It makes sense from the picture: to
grow an n \times n square by one, you add a strip down one side, along
the top, and one corner — that's 2n + 1 extra unit squares, always odd.)
Squares and cubes are also the first two rungs of the power ladder —
n^2, n^3, then n^4
and beyond — which grows into the whole toolkit of
the laws of indices.
And here's the scandal. The ancient Greek followers of Pythagoras believed every
number could be written as a neat fraction. Then someone proved that
\sqrt{2} — the diagonal of a simple unit square — is
irrational and can't be a fraction at all. Legend says the discovery was so shocking, so
threatening to their worldview, that it was kept a guarded secret. Not all numbers, it turned
out, are tidy.
See it explained