Powers, Exponents, and Roots

Squares, Cubes, and Higher Order Exponents

Summary Squares, Cubes, and Higher Order Exponents

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.