In mathematics, exponentiation is the name for the operation also called raising to a power. In simple cases, it refers to repeated multiplication. It is indicated by a superscript, a small number or expression written above the line:
- 26, "two to the sixth power," 2 · 2 · 2 · 2 · 2 · 2 = 64
- x4, "x to the fourth power," x · x · x · x
- yn, "y to the n-th power," y · y · y · ... · y, where y appears n times.
The superscripted value is called the exponent. The definition of exponentiation leads to the notion of exponential functions, where the exponent becomes a variable.
The definition of exponentiation as repeated multiplication only makes sense when the exponent is a positive integer—what does it mean to say "x multiplied by itself half a time" or "minus three times?" However, mathematicians have found logical meanings for zero, negative, fractional, and even complex exponents. These meanings arise from the basic observation that
- (xa) · (xb) = x(a + b)
We can show that the zeroth power of any nonzero number is 1
- (x0) · (xn) = x0 + n = xn
Dividing both sides by xn we get
- (x0) = 1
A negative exponent produces the reciprocal of the corresponding positive exponent:
- (x-n) · (xn) = xn - n = x0 = 1
Dividing both sizes by (xn) we get
- (x-n) = 1 / xn
Fractional exponents 1/n give us the n-th root; for example, x0.5 gives us the square root:
- (x0.5) · (x0.5) = x1 = x
A geometric sequence can be used to represent compound interest, exponential growth, or exponential decay. Because of the compounding effect, exponential growth occurs faster and faster, like wildfire. This has led to the colloquial use of the word exponential to mean "growing very rapidly," or even "very large."