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

Rules and properties of radicals and/or fractional exponents

Operations on radical expressions

Radicals and/or fractional (rational) exponents
Radicals (roots) and/or exponentiation with fractional (rational) exponent
 The nth root of a real number a, denoted is defined to be such number x that  xn = a,
 or 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 radical.
 The symbol is called the radical sign.
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.
Rules and properties of radicals
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
As, to simplify means to reduce given expression to a simpler form, or find another expression with the same value. So, for example a square root radical expression is simplified when its radicand has no square factors.
To simplify a radical means to remove factors from the radical until no factor in the radicand has an exponent greater than or equal to the index of the radical and the index is as low as possible.
Therefore, use the product rule of radicals to simplify the nth root radicals. Throughout, we take variables to be positive, unless the index of the radical is odd.
 Examples:
Operations on radical expressions
Throughout, we take variables to be positive, unless the index of the radical is odd.