Radicals (roots) and/or exponentiation with fractional (rational) exponent
      Rules and properties of radicals
      Rules and properties of radicals and/or fractional exponents
      Simplifying radical expressions
      Operations on radical expressions
         Adding, subtracting and multiplying radicals
      Rationalizing a denominator
      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 
1. 
2. 
3. 
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
   
   
   
   
   
   
   
   
Simplifying radical expressions
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.
Adding, subtracting and multiplying radicals
Examples:
 
Examples:
                 
Examples:
Examples:
 
Examples:
 
Rationalizing a denominator
Rationalizing a denominator is a method for changing an irrational denominator into a rational one.
Examples:
 
        since   a3 - b3 = (a - b) (a2 + ab + b2)
Radicals and/or fractional (rational) exponents
Examples:
 
Examples:
 
Intermediate algebra contents
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