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Radicals
Radicals
The symbol
a
is a radical.
The positive number a is the radicand.
We say the square root of a or root a.
The square root of a number n
is the number which, when
squared is n.
 n = n
2
For example :
5 and -5 are both the square root of 25,
2
2
since 5 = 25 and (-5) = 25.
√ 25 = 5 and -5
The symbols:
indicates the positive root
5 2 = 25
-
25 = 5
indicates the negative root
-5 2 = 25

then
then -
25 = -5
indicates both roots

25 = 5 and -5
Some examples:
8
- 36 = – 6
 4 = ±2
64 =
Why can’t we find a square root for 2
36... in other words can b = -36?
62 = 36
(-6)2 = 36
6(-6) = -36
The square of a number can never be
negative.
Therefore, the square root of a
negative number does not exist in the
real numbers.
Radical Rules:
Product Rule-The square root of a product
is equal to the the product of
the square roots.
ab =
a •
b
Quotient Rule-The square root of a quotient
is equal to the quotient of the
two radicals.
a
b
a
=
b
Important Products:
 n
2
 n •  n 
n2
n is positive
=n
Radicals are simplified when:
1) the radicand has no perfect
square factors
2) the denominator of a fraction is
never under a radical.
The product and quotient rules allow
radical expressions to be simplified.
#1 “Take out” perfect square factors
Rewrite the radicand as a product of
its factors; with the largest perfect
square factor possible. Use the
product rule to simplify the root of
the perfect square as a rational
number, leaving the other factor
under the radical.
#2 Rationalize the denominator:
To create a fraction with a rational
denominator, multiply both
numerator and denominator by the
the irrational number found in the
denominator of the fraction.
Examples
Comparison with variables