Squares
The square of a number is that number times itself. 5 squared, denoted
52, is equal to 5×5, or 25. 2 squared is 22 = 2×2 = 4.
One way to remember the term "square" is that there are two dimensions in a
square (height and width) and the number being squared appears twice in
the calculation. In fact, the term "square" is no coincidence--the square of a
number is the area of the square with sides equal to that number.
A number that is the square of a whole number is called a perfect square.
42 = 16, so 16 is a perfect square. 25 and 4 are also perfect
squares. We can list the perfect squares in order, starting with 12:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...
Cubes
The cube of a number is that number times itself times itself. 5 cubed,
denoted 53, is equal to 5×5×5, or 125. 2 cubed is
23 = 2×2×2 = 8. The term "cube" can be remembered
because there are three dimensions in a cube (height, width, and depth)
and the number being cubed appears three times in the calculation.
Similar to the square, the cube of a number is the volume of the cube
with sides equal to that number--this will come in handy in higher levels of math.
Exponents
The "2" in "52" and the "3" in "53" are called
exponents. An exponent indicates the number of times we must multiply the
base number. To compute 52, we multiply 5 two times (5×5), and
to compute 53, we multiply 5 three times (5×5×5).
Exponents can be greater than 2 or 3. In fact, an exponent can be any number.
We write an expression such as "74" and say "seven to the fourth
power." Similarly, 59 is "five to the ninth power," and
1156 is "eleven to the fifty-sixth power."
Since any number times zero is zero, zero to any (positive) power is always
zero. For example, 031 = 0.
