Powers, Exponents, and Roots

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.