Fractional exponent is a natural extension to the integer exponent.
We already know, that if b is positive integer, then
- ab=ba⋅a⋅a⋅...⋅a (see positive exponent)
- a−b=ab1=ba⋅a⋅a⋅...⋅a1 (see negative exponets)
But what if exponent is a fraction?
To understand addition of exponents, let's start from a simple example.
Example. Suppose, we want to find 23⋅24.
We already learned about positive integer exponets, so we can write, that 23=2⋅2⋅2 and 24=2⋅2⋅2⋅2.
To understand subtraction of exponents, let's start from a simple example.
Example. Suppose, we want to find 2427.
We already learned about positive integer exponets, so we can write, that 27=2⋅2⋅2⋅2⋅2⋅2⋅2 and 24=2⋅2⋅2⋅2.
To understand multiplication of exponents, let's start from a simple example.
Example. Suppose, we want to find (23)4.
We already learned about positive integer exponets, so we can rewrite outer exponent: (23)4=23⋅23⋅23⋅23.
We already saw division of exponents two times:
- when discussed fractional exponents (anm=nam)
- when discussed multiplication of exponents (indeed, anm=am⋅n1=(am)n1=nam).
Rule for dividing exponents: nam=anm.
Properties (rules) of exponents:
- Zero power: a0=1, a=0
- Zero base: 0a=0, a=0
- 00 is undefined
- 1a=1
- Negative exponent: a−b=ab1, b=0
- Nth root: an1=na, n=0
- Addition of exponents: am⋅an=am+n
- Subtraction of exponents: anam=am−n, a=0
- Multiplication of exponents: (am)n=am⋅n=(an)m
- Division of exponents: nam=anm, n=0
- mam=a, if m is odd
- mam=∣a∣, if m is even
- nam=(na)m (just pay attention to signs and check, whether number exists)
- Power of a product: an⋅bn=(ab)n
- Power of a quotient: bnan=(ba)n, b=0
We already covered all rules earlier, except last two.