For any positive integer n, exponentiation is defined by:
The factor x can be in the set of complex numbers. Examples,xn = x·x·x·...·x (n factors of x)
Consider: (2×2×2)×(2×2) = 23×22 = 25 = 32. The exponents add. This gives us the fundamental law of exponentiation:x4 = x·x·x·x 5x4yz3 = 5x·x·x·x·y·z·z·z 4a3b2c = 4a·a·a·b·b·c 1.13 = 1.331 (1 + i)2 = 1 + 2i + i2 = 2i
Now suppose that one exponent is zero. For example, 20×22 = 22; then 20 must be equal to one. So, in general, the identity x0 = 1 is true. However, if x = 0, this would mean that 00 = 1 which is a problem since 0n must equal zero. Therefore, the term 00 cannot be defined. We can write the basic law for a zero exponent as:xm+n = xmxn
What if an exponent is negative? The equality 2-2×22 = 20 satisfies the fundamental law given above. This implies that the factor 2-2 must be equal to 1/22. That is, a negative exponential is defined as the reciprocal of a positive exponential. This basic law can be stated as:x0 = 1 for all non-zero x
Note that, 0-n is not defined for any positive integer n because raising 0 to a negative power would imply division by 0. Now for some examples:x-n = 1/xn for all non-zero x
What about products of exponentials? If we group a repeated multiplication as follows,x-4 = 1/x4 3y-4 = 3/y4 4-3 = 1/64 -5-4 = -1/625 (exponentiation is done before multiplication) 3x-2 + 4x-1 + 2 = (3 + 4x + 2x2)/x2 (1 + i)-2 = 1/(1 + i)2 = 1/(2i) = 0.5i-1 = -0.5i
Then we see that the exponents multiply. Then, another basic law of exponents is:(2×2)×(2×2)×(2×2) = (2×2)3 = (22)3 = 26
Other laws can be derived from these basic laws. The following summarizes the laws that apply to integer exponents:(xm)n = xm·n
Same base, different exponents
xmxn = xm+n
xm/xn = xm-n, x ≠ 0
x-n = 1/xn = (1/x)n
(xm)n = xmn
x0 = 1, x ≠ 0
x-1 = 1/x, x ≠ 0
if (n = m) and (x ≠ 0) and (x ≠ 1) and (x ≠ -1) then
xn = xm
Same exponent, different bases
(xy)n = xnyn
(x/y)n = xn/yn, y ≠ 0
(x/y)-n = yn/xn
if (x = y) and (x > 0) and (y > 0) and (n ≠ 0) then
xn = yn