Squares
The square of a number is that number times itself. 5 squared, denoted
5^{2}, is equal to 5×5, or 25. 2 squared is 2^{2} = 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.
4^{2} = 16, so 16 is a perfect square. 25 and 4 are also perfect
squares. We can list the perfect squares in order, starting with 1^{2}:
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 5^{3}, is equal to 5×5×5, or 125. 2 cubed is
2^{3} = 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 "5^{2}" and the "3" in "5^{3}" are called
exponents. An exponent indicates the number of times we must multiply the
base number. To compute 5^{2}, we multiply 5 two times (5×5), and
to compute 5^{3}, 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 "7^{4}" and say "seven to the fourth
power." Similarly, 5^{9} is "five to the ninth power," and
11^{56} is "eleven to the fifty-sixth power."
Since any number times zero is zero, zero to any (positive) power is always
zero. For example, 0^{31} = 0.