Download Chapter 5, Section 6

Chapter 5, Section 6
Radical Expressions
Radical Expressions
Radical expressions are arithmetic
expressions that contain a radical sign.
These expressions may be simplified,
added, subtracted, multiplied, or divided.
Simplifying Radical Expressions
A radical expression is simplified if:
• the radicand contains only powers LESS
THAN the index
• the radicand contains no fractions
• no radicals appear in the denominator
Example 1
Simplify
4 9
25a b
First we’ll want to factor 25 into 52.
Next, we’ll want to remove any number of
factors equal to the index. When dividing
by the index, any remainder “remains”
under the radical:
2
52 a4b9  2 52  2 a4  2 b9  5  a 2  b4 2 b  5a 2b4 b
Quotient Property of Radicals
Recall that for n>1, and any a and b, where
roots are defined and b ≠ 0,
n
a na
n
b
b
Rationalizing the denominator
Remember that there can be no radicals in the
denominator, so if while we are working we GET
a radical in the denominator, we get rid of it by
MULTIPLYING by just enough factors of it
(under a radical of the same index) to remove it,
i.e. to make the exponent equal the index.
We just need to make sure we multiply the
numerator by the same number.
Example 2
8
y
x7
Simplify
We will apply the Quotient Property of Radicals
and rationalize the denominator:
8
2
y

7
x
2
2
y
x
8
7

y
x
3
4
x

y
x
3
4
x


x x
y
Note that we don’t care if the radical is in the
NUMERATOR…
4
x4
x
Example 3
Simplify
3
2
9x
We will again apply the Quotient Property of
Radicals and rationalize the denominator:
3
2

9x
3
3
2

9x
3
3
2
32 x1

3
3
2
2 1
3 x

3
3x
2
3
1
2
3x

3
6x
3x
2
Product Property of Radicals
When multiplying two radicals with the same
index, just multiply the radicands:
n
a  b  ab
n
n
Remember that by the Symmetric Property
of Equality, the reverse is true:
n
ab  n a  n b
Example 4
Simplify 5 100a  73 10a
We will use the Commutative and
Associative properties, then apply the
Product Property of Radicals. Remember
to always factor any coefficients.
2
3
5 100a  7 10a  5  7  100a  3 10a 
3
2
3
2
3
35  1000a  35  10 a  35 10a  350a
3
3
3
3
3
Adding and Subtracting Radicals
Remember that we can only add and
subtract like quantities, so we can only
add and subtract the same radicals (same
index, same radicand). E.g.
35 7  55 7  85 7
We MUST simplify all radicals before we can
determine if they can be added or
subtracted, however.
Example 5
Simplify 3 45  5 80  4 20
Even though these radicals are not the
same (their radicands are different), we
cannot tell if they can be combined until
we simplify:
3 45  5 80  4 20 
3 32  5  5 24  5  4 22  5 
9 5  20 5  8 5   3 5
Multiplying and Dividing Radical
Expressions
You know how to FOIL (i.e. you know how to apply
the Distributive Property!), so multiplication won’t
be much of a problem.
Division presents an added wrinkle because we
NEVER leave radicals in the denominator.
To get rid of binomials with radical expressions, we
will remember that (a + b)(a - b) = a2 - b2.
The numbers a + b and a - b are called
conjugates of each other, and when you have a
radical expression of the form a ± b in the
denominator, you can get rid of it by multiplying
by the conjugate of the expression.
Example 6
Simplify (2 3  3 5 )(3  3 )
We see this is a binomial times a binomial,
so we will FOIL it:
(2 3  3 5 )(3  3 ) 
(2 3 )(3)  (2 3 )(  3 )  (3 5 )(3)  (3 5 )(  3 ) 
6 3  6  9 5  3 15
Example 7
2 3
Simplify
4 3
This looks pretty simple already, but it has a
radical in the denominator. We will
multiply the denominator (and numerator!)
by its conjugate to clear out the radical:
2  3 2  3 4  3 8  2 3  4 3  3 11 6 3




2
2
13
4  ( 3)
4 3 4 3 4 3
Your homework…
Chapter 5 section 6, 15-47 odd, 59-81 odd,
PLUS 20, 30, 34, 48, and 66.
If you pursue evil with pleasure, the pleasure
passes away and the evil remains; if you
pursue good with labor, the labor passes
away but the good remains.
--Cicero