To extend exponentiation to the use rational numbers as exponents, we define the principle nth root of x when n is a positive odd integer greater than 1, as:
_
x1/n = n√x
In which the symbol √ is called a radical, n is the index, and
x is the radicand. If n is omitted, the index is equal to 2.
When x is a real number, the radical term has the unique value y
such that:
When n is a positive even number, the principle nth root of x is defined as:_ (n√x)n = (x1/n)n = x
_
x1/n = n√x if x > 0
x1/n = 0 if x = 0
x1/n is not a real number if x < 0
The following laws apply to principal roots:
Some examples,(xn)1/n = x if n is odd or if n is even and x ≥ 0 (xn)1/n = |x| if n is even and x < 0
Exponentiation with a fractional exponent is defined as integer exponentiation of the principle root:81/3 = 2 (-8)1/3 = -2 321/5 = 2
_ _
xm/n = (n√x)m = n√xm
For example,
___ __
82/3 = 3√(82) = 3√64 = 4
1252/3 = (1251/3)2 = 52 = 25
(-8)-4/3 = 645/6 = 32
Radical have properties that are consistent with those of exponents. These can all
be derived from the laws for exponents:
Often it is desirable to reduce a radical to its simplest form. There are several accepted conventions:(n√x)n = x n√(xn) = x if x ≥ 0 n√(xn) = x if n is odd and x < 0 n√(xn) = |x| if n is even and x < 0 n√(ab) = n√a n√a n√(a/b) = n√a/n√a n√m√x = mn√x
3√(x4) = x 3√x
8√(x2) = 4·2√(x2) = 4√√(x2) = 4√x
3 3√2 --- = --- √2 2
√2 √6
√(2/3) = --- = ---
√3 3
So, 41/2 can be equal to +2 or -2. That is, it has two roots.(-2)2 = 4 (+2)2 = 4