Exponents are used to denote the repeated multiplication of a number by itself; for example, \((5)^4 = 5 * 5 * 5 * 5 = 625\). In the expression 5 is called the base, 4 is called the exponent, and we read the expression as “5 to the fourth power.” So 5 to the fourth power is 625. When the exponent is 2, we call the process squaring. Thus 8 squared is 64.

When negative numbers are raised to powers, the result may be positive or negative. For example, \((-2)^2 = (-2) * (-2) = 4\) and \((-2)^3 = (-2) * (-2) * (-2) = -8\). A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative. Note that without the parentheses, the expression means “the negative of ‘3 squared’ ”; that is, the exponent is applied before the negative sign.




Exponents can also be negative or zero; such exponents are defined as follows.

• For all nonzero numbers a, \(a ^ 0 = 1\). The expression \(0 ^ 0\) is undefined.
  
• For all nonzero numbers a, \(a ^ -^1 = \frac{1}{a}\), \(a ^ -^2 = \frac{1}{a^2}\)