Exponentiation

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 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

• (x^a) · (x^b) = x^{(a + b)}

We can show that the zeroth power of any nonzero number is 1

• (x⁰) · (xⁿ) = x^{0 + 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) = x^{n - n} = x⁰ = 1

Dividing both sizes by (xⁿ) we get

• (x^-n) = 1 / xⁿ

Fractional exponents give us surds; for example, x^0.5 gives us the square root:

• (x^0.5) · (x^0.5) = x¹ = x