# Power

In math, the terms "power" and "exponent" are often used interchangeably to refer to "n" in the expression bn. This expression can be read as "b to the power of n." The term power can also refer to the result of the expression.

52 = 25

In the above expression, both 2 and 25 may be referred to as a power, though the latter is less common.

## Common powers

The two most common powers used in math are squares and cubes.

### Squares

Squaring a base refers to multiplying the base by itself. It is the same as raising a base to the power of 2. The square of a number can be referred to as a perfect square.

72 = 7 × 7 = 49

The above expression is most commonly read as "seven squared," but can also be read as "seven to the power of two," or "seven to the second power." The result, 49, is the perfect square of 7.

One important property of squaring is that the square of an expression is equal to the square of its additive inverse. For example 72 = (-7)2 = 49. This is also true for algebraic expressions.

Example

Show that the square of (x - 1) is equal to the square of its additive inverse, (-x + 1).

 (x - 1)2 = (x - 1)(x - 1) = (x)(x) + (x)(-1) + (-1)(x) + (-1)(-1) = x2 - x - x + 1 = x2 - 2x + 1 (-x + 1)2 = (-x + 1)(-x + 1) = (-x)(-x) + (-x)(1) + (1)(-x) + (1)(1) = x2 - x - x + 1 = x2 - 2x + 1

### Cubes

Cubing a base means multiplying it by itself two additional times; given a base b, b cubed is b × b × b. The cube of an expression is the same as raising it to the third power. Given b = 5:

b3 = 5 × 5 × 5 = 125

The result of cubing an integer is a perfect cube. In the example above, 125 is the perfect cube of 5.

The name "cube" comes from the fact that the volume of a geometric cube can be found by cubing its side length, similar to how squaring the side lengths of a square yields its area. Consequently, multiplying a number, n, by its square, gives us its cube.

## Basic power properties

If the power is a positive integer, then the power tells us how many times to multiply the base by itself.

52 = 5 × 5 = 25

If the power is a negative integer, -n, and b is a non-zero real number, we take the reciprocal of the base raised to the power of n.

b-n = 5-2 = Anything raised to the power of 0 is 1. 00 is a special case that is either considered 1 or undefined.

1360 = 1

If the power is a fraction, we take the nth root of the base, where n is the number in the denominator of the fraction. 