Let b be a real number. In the expression br, b is called the base and r is called the exponent.

(1) Natural Powers

  If r=n where n is a positive integer, thenbn=bbbn times


(2) Zero Power

If r= 0, we define

b0=1if b0

00 is not defined.


(3) Rational Powers

If r=1/n where n is a positive integer, then b1/n (also denoted by bn), called the nth root of b, is a real number u such that un=b. bn=b1/n=umeansun=b

We need to consider two cases.

    • Case 1:  n is a positive odd integer. There is exactly one nth root for each real b.
    • Case 2: n is a positive even integer:

Case 2a: If  b < 0 , there is no nth root. 

[If we multiply a positive or negative number by itself an even number of times the result will be positive (for example, (-2)4=32>0). So there cannot be a real number u such that un=b as un ≥ 0 for all real u and positive even integer n.

Case 2b: If b > 0, there are two nth roots (one positive and one negative). 

However, the symbol  b1/n or bn is reserved for the positive nth root.
[There are two nth roots because (-u)n=un when n is even; that is, for each b>0 there will be two nth roots, u and -u.]  

Remark:

    • bn>0 if b>0
    • bn<0 if b<0 and n is odd
    • bn is imaginary ( = not real) if b<0 and n is even.

 

Properties of the nth roots

Let a and b be two real numbers and m and n be two integers. We can show that the nth roots have the following properties.

(a) abn=anbn             (equivalently (ab)1/n=a1/nb1/n)

(b) abn=anbn             (equivalently  (ab)1/n=a1/nb1/n)

(c) anm=amn          (equivalently (a1/n)1/m=a1/(mn))

(d) (an)1/n=ann={a(if n is odd)|a|(if n is even)

  • Note that when n is even and both a<0 and b<0, then abn and a/bn exist (they are real numbers) because ab>0 and a/b>0, but an and bn do not exist (in fact, they are imaginary).

 

If r=m/n where m and n are positive integers and m/n is in lowest terms, bm/n is defined to be (b1/n)m; that is, the nth root of the mth power of b.

bm/n=(bm)1/n

We can verify that also bm/n=(b1/n)m.

If n is even, we require that b > 0.


(4) Irrational Powers

If r=α where α is an irrational number and b>0, the approximate value of bα is obtained by expressing α approximately as a fraction. For example, to calculate b2 for any given b > 0, we can replace 2 by enough digits of its decimal expansion. For instance, 3231.41 and with better accuracy 3231.4142.

Because we can write 1.41 as 141/100, 31.41 is raising 3 to a fraction m/n discussed above; the same applies to 31.4142.

  • Note that when α is irrational, bα is imaginary for b < 0 .

(5) Negative Powers

For any real number r, if b 0, we define br to be 1br whenever br is defined.

br=1br

For example, 

b12=1b1/2=1b,b53=1b5/3=1b53=1b3b23=1b33b23=1bb23


General Properties

If b>0, c>0, and r and s are two real numbers, we can prove: 

(a)    brbs=br+s

(b)    brbs=brs

(c)    (br)s=brs

(d)    (bc)r=brcr

(e)    (bc)r=brcr