Radicals
Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

 

A radical
    or the principal nth root of
k:
     k,
the radicand, is a real number.
     n,
the index, is a positive integer greater than one.
                        

Properties of Radicals:

Simplifying Radicals:
Radicals that are simplified have:
 - no fractions left under the radical.
 - no perfect power factors in the radicand, k.
 - no exponents in the radicand, k, greater than
   the index, n.
 - no radicals appearing in the denominator
   of a fractional answer.

 

Examples:  (The following examples demonstrate various solution methods.)

Simplify:  Factor the radicand to isolate the perfect power factor(s), which will allow them to be removed from under the radical.  You will need to remember your rules for working with exponents in order to isolate the perfect powers.
perfect squares
  4, 9, 16, 25, 36, ...
  x2, x4, x6, x8, ...
  x2y2, x2y4, 16x6y8, ...
powers are "even"
perfect cubes
  8, 27, 64, 125, ...
  x3, x6, x9, x12, ...
  x3y3, x3y6, 27x6y9, ...
powers are "multiples of 3"

1.    (notice the perfect cube of 8 being isolated)

2. 
       (in this problem, several perfect squares were isolated, namely 9, a2 and b4)

3.      (the -8 and the y3 are perfect cubes)

4.  (x5 is a perfect power of 5)

 

To add radicals, simplify first if possible, and add "like" radicals.

1.               2.  

   (After simplifying the radicals, it became                        (Be sure to combine ONLY like radicals.  Also
     apparent which radicals could be added.)                     be sure to continue writing the index of 3.  Failing
                                                                                        to do so, creates an incorrect answer using square
                                                                                        root, instead of the correct cube root.)

To multiply radicals apply and/or distributive multiplication (FOIL).

1.    (multiply outside, multiply inside, and simplify)

2.  (notice the perfect powers isolated)

3.     (distribute across the parentheses)

4. 
    
(this problem requires multiplication of two binomials by distributive multiplication or FOIL.  Notice how
        none of the terms can be combined in the final answer.)

5. 
    
(again binomial multiplication is needed.  This time, however, terms can be combined for the final answer.)

6. 
    
(again binomial multiplication is needed.  The two terms being multiplied in this problem are conjugates.
        Notice how the middle terms of the answer drop out.)


Division of radicals will be discussed in
Rationalizing Denominators with Radicals.