Radicals (roots) and/or exponentiation with fractional (rational) exponent

Rules and properties of radicals and/or fractional exponents

Radicals (roots) and/or exponentiation with fractional (rational) exponent
 The nth root of a real number a, denoted is defined to be that number x such that  xn = a, i.e.,
 where n is a natural number greater than 1.
 Thus,  if a = 0,  then
 if a > 0,  then is the positive real number x such that  xn = a,
 if a < 0 and n is odd, then is the negative real number x such that  xn = a,
 if a < 0 and n is even, then is not a real number,
if n is an even number, the nth root of a is defined to be that number x 0 such that xn = a.
Even roots exist only for nonnegative numbers.
 Odd roots always exist and have the property that where n is odd.
 The expression is called a radical, the number a is called the radicand, and n is the index of the
The principal nth root of a real number a is the unique real number x which is an nth root of a and is of the same sign as a.
Let n be a natural number and a be a real number, then
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 Examples: 1. The odd root of any real number exists:
 a) since  33 = 27. b) since  (-3)3 = (-3)·(-3)·(-3) = - 27. c) since  (-2)5 = (-2)·(-2)·(-2)·(-2)·(-2) = -32.
 2. The even root of a nonnegative real number exists:
 a) since  42 = 16 and  4 > 0,   [recall that  ( ± 4)2 = 16]. b) since  34 = 81 and  3 > 0,   [recall that  ( ± 3)4 = 81].
 Note that the even root is defined to be a nonnegative number.
 3. The even root of a negative real number does not exist as a real number.
 For example, do not exist as real numbers, but they do exist as complex numbers.
Rules and properties of radicals and/or fractional exponents
 If m, n and p are natural numbers (n >1) and if  a and b are nonnegative real numbers, then Properties Examples