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 number has just one real nth root when n is odd (odd index): odd
roots of positive numbers are positive and nth roots of negative numbers or 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 nth roots when n 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
Set radicand  0 and
Must be non-negative   .
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 
. If any even index is in place, the same principle applies.
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
If the radicand contains a variable, be sure to include absolute value bars if needed, to
indicate only the principal root is the solution!
EX: If x is any real number, then 4 x 2  2 x . Watch for directions that specify
that no radicands were formed by raising a negative number to an even power:
then absolute value signs are not required.
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
 3 33  5  x 5 y 13
EX:
 3 33  5  x 3 x 2 y 12 y
 
3
 3 33  5  x 3 x 2 y 4 y
 3xy 4 3 5 x 2 y
PRODUCT RULE OF RADICALS: n ab  n a  n b . This rule is used for simplifying
(as shown above), as well as for multiplying
3x  12 x 2  36 x 3  6 2 x 2 x  6 x x
EX:
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  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. Be sure to simplify
radical expressions before attempting to add or subtract.
9x  9  x  1
EX:
 9( x  1)  x  1
 3 x 1  x 1
 4 x 1
RATIONALIZING THE DENOMINATOR: Multiply both numerator and
denominator by factors that will allow denominator radicand to contain only perfect “n”
factors (where n is the index)
EX:
3
5x
5x
2 2  3  x 3 2 2  3  5  x 2 3 150 x 2
3




6x
18 x 2
2  32  x 2 2 2  3  x
2 3  33  x 3
PRINCIPLE OF POWERS: If a  b , then a n  b n for any exponent n.
PRINCIPLE OF SQUARE ROOTS: If x 2  n , then x  n or x   n
IMAGINARY NUMBERS:
1  i , and i 2  1
PYTHAGOREAN THEOREM: In any right triangle where a and b are the length of
the legs and c is the length of the hypotenuse, a 2  b 2  c 2
c
a
b