Simplifying
When simplifying a radical
expression, find the factors that
are to the nth powers of the
radicand and then use the
Product Property of Radicals.
What is the Product
Property of Radicals???
Product Property of Radicals
For any real numbers a and b,
and any integer n, n>1,
1. If n is even, then ab  a  b
n
n
n
When a and b are both nonnegative.
2. If n is odd, then
n
ab  n a  n b
Let’s do a few problems together.
8
1.
144x y
2
5
4 2
2 2
12  (x ) (y )  y
2
4 2
2 2
12  (x )  (y )  y
4
12x y
2
y
Factor into
squares
Product
Property of
Radicals
3
2
2) 7 64n  4 8n 
3
3
Product Property
of Radicals
2
7  4  64n 8n
3
3
28  (4)  (2)  n
3
3
3
3
3
28  (4)  (2)  n
3
3
28  4 2  n  224n
3
Factor into
cubes if possible
Product Property
of Radicals
Now, you try these examples.
1)
40x
6
2)
3)
9 11
56a b c
3
3 5
54x y
16x y  3xy
2
4)
5)
3
5
3
27x  9x y
3
5
2 6
Quotient Property of Radicals
For real numbers a and b, b  0,
And any integer n, n>1,
n
n
Ex:
a
a
 n , if all roots are defined.
b
b
81

256
81
9

256 16
In general, a radical expression is
simplified when:
The radicand contains no fractions.
No radicals appear in the
denominator.(Rationalization)
The radicand contains no factors
that are nth powers of an integer
or polynomial.
Simplify each expression.
3
6
1)
6
x

y y
x
3
y
x

3
y
3
3
x
x

2
y
y
x3

y y
y Rationalize

y the
denominator
3
x y

yy
3
x y
2
y
Answer
2)
5
5
5
4 
4x
5
4 
4x


5
5
5
5
4
4x
5
40x
5
5
2 x
5
40x
2x
5
8x
5
8x
5
Why use
5
2
8x
?
8x
3
4 = 2 , 8= 2
To simplify a radical by adding
or subtracting you must have
like terms.
Like terms are when the powers
AND radicand are the same.
Ex:
3
3
5 and 6 5, 2x 6z and 5x
6z
Here is an example that we will do
together.
3 20  150  5 45
2
2
2
3 2  5  5  6  5 3 5
Rewrite
using factors
3 2 5  5 6  5 3 5
6 5  5 6 15 5
9 5  5 6
Combine like terms
Try this one on your own.
4 3  5 12  7 27
You can add or subtract radicals
like monomials. You can also
simplify radicals by using the
FOIL method of multiplying
binomials.
Let us try one.
Ex: (3 6  2 3)(4  3)
(3 6  2 3)(4  3)
F
O
I
L
3 6 4  3 6 3  2 3  4  2 3  3
2
12 6  3 3  2  8 3  2 3
12 6  9 2  8 3  6
Since there are no like terms, you
can not combine.
Lets do another one.
(8  5 3)(8  5 3)
8 8  8 5 3  5 3 8  5 3 5 3
64  40 3  40 3  25 3
64  75
11
When there is a binomial with a
radical in the denominator of a
fraction, you find the conjugate and
multiply. This gives a rational
denominator.
Ex:
5  6  Conjugate:
56
3  2 2  Conjugate: 3  2 2
What is conjugate of
2 7  3?
Answer: 2 7  3
Simplify:
56
53
56
56 53


53
53 53
5  3 5  6 5  18
59
Next
Multiply by
the conjugate.
FOIL numerator
and denominator.
23  9 5
4
Combine like terms
Try this on your own:
6
3 2
3 62 3
Answer:
7
Here are a mixed set of problems
to do.
1)
2)
3)
4)
540
3
6 (43 12  53 9 )
8
3
9x
4
4
120  4 30
5) ( 7  2 2)( 6  2 2 )
3 5
6)
4 3