Download Important Radical Information

Important Radical Information
By definition, an nth root of a number is this:
The number c is an nth root of a if c n  a .
EX: The square roots of 81 are 9 and -9, because 9 2  81 and (9) 2  81
The index of a radical symbol n
EX:
3
  indicates what root is being found.
n
x means the “cube root” of x. Similarly,
7
x means the “7th root” of x.
ODD ROOTS: Every real number has just one real odd root (index is odd): odd roots of
positive numbers are positive and odd roots of negative numbers are negative.
EX: 5 32 = 2 and 5  32  2 . It’s not a problem to have a negative radicand with an odd
root.
EVEN ROOTS: Every positive real number has two real even roots (index is even).
Negative numbers do not have real nth roots when n is even.
EX: The 4th roots of 16 are 2 and -2. Negative 16 (-16) does NOT have a 4th root, since no
real number multiplied to itself 4 times can be negative.
The chart below summarizes the rules for radicands, depending on the index, and how to
find the domain of a radical function. REMEMBER: the domain rules only apply if the
function does not contain the radical in any denominator.
Index
Radicand possibilities
To find domain:
Even
Must be non-negative  0 . Set radicand  0 and
solve for the variable
The even root of a negative
number does not exist
Odd
Can be any real number
Domain can be any
real number 
Any positive number has two square roots. However, the
symbol indicates the
“principal,” or positive square root of a number only. The secondary (negative) root is
indicated by the symbol 
. The same principal applies for any even index.
EX: Even though 16 had two square roots (4 and -4), 16 ONLY equals 4, NOT -4. If I
want to refer to the negative root (-4), I would need to say  16  4
SIMPLIFYING RADICALS involves removing perfect nth power factors from the
radicand. The best way to do this is to factor the radicand in such a way that all factors in
the radicand are written as a perfect nth power (n or a multiple of n), or as something to a
power less than the index n.
SIMPLIFY
3
135 x 5 y 10
Factor 135 into primes
 3 33  5  x 5 y 13
EXAMPLE:
Rewrite variable factors either with exponents that are multiples
of the index (3) or with exponents less than the index (3)
 3 33  5  x 3 x 2 y 12 y
 3 33  x 3 y 12  3 5 x 2 y
Put all perfect cube factors under one radical, and all other factors
under a separate radical
 3 xy 4 3 5 x 2 y
Simplify the radical containing perfect cubes.
PRODUCT RULE OF RADICALS: n ab  n a  n b . This rule is used for simplifying (as
shown above), as well as for multiplying
EX:
3x  12 x 2  36 x 3  6 2 x 2 x  6 2 x 2  x  6 x x
QUOTIENT RULE OF RADICALS:
EX:
2 150 xy
3x
 2
n
a

b
n
a
n
b
150 xy
 2  50 y  2  25  2 y  2  25  2 y  2  5 2 y  10 2 y
3x
ADDING AND SUBTRACTING: Only “like radical terms” (ones in which the radicand
and the index are identical) can be added or subtracted. This concept is very similar to the
idea of adding and subtracting like terms. When added, the radical will be the SAME as it
was in the like radical terms. Be sure to simplify radical expressions before attempting to
add or subtract. Remember about invisible 1’s in front of radicals if no other “coefficient”
is showing.
9x  9  x  1
EX:
 9( x  1)  x  1
 3 x 1  x 1
Invisible “1” in front of second radical.
 4 x 1
PYTHAGOREAN THEOREM: For any right triangle with legs a and b and hypotenuse c,
a 2  b2  c2
c
a
b