Squares, Cubes and Roots
Squares and cubes
A square number is a number multiplied by itself —
n \times n = n^2 (read “n squared”). It is the
area of a
square with side n:
1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100,\ \dots
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:
1,\ 8,\ 27,\ 64,\ 125,\ \dots
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
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a square is n^2 = n \times n; a
cube is n^3 = n \times n \times n;
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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;
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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.