Transcript Simplifying Radical Expressions

Simplifying Radical
Expressions
For a radical expression to be simplified it has to satisfy
the following conditions:
1.
2.
3.
4.
The radicand has no factor raised to a power greater than or
equal to the index. (EX:There are no perfect-square factors.)
The radicand has no fractions.
No denominator contains a radical.
Exponents in the radicand and the index of the radical have no
common factor, other than one.
Converting roots into fractional
exponents:
 Any radical expression For example
may be transformed
into an expression with
a fractional exponent.
The key is to remember
that the fractional
exponent must be in the
form power
root
16 
5
3
 2    2
1
16
=
2
5
3
Negative Exponents:
 Remember that a
negative in the
exponent does not
make the number
negative!
 If a base has a negative
exponent, that indicates
it is in the “wrong”
position in fraction. That
base can be moved
across the fraction bar
and given a postive
exponent.
EXAMPLES:
2
2
x
x
 2 
1
2
4
x 2
1
1
1
81  1  4 
81 4
81 3
3
1
1
1
4
16  3 

16 4 4 16 3 8
1
4
Simplifying Radicals by using the
Product Rule
m
m
a
&
b are real
 If
numbers and m is a
natural number, then
m
Examples:
3
20 
a  b  ab
m
m
So, the product of two
radicals is the radical of
their product!
733 
3
7 3 
45 
3
21
4 5 2 5
6
10m3  6 5m 2  6 50m5
3
7 5 
3
7 5
*This one can not be simplified
any further due to their indexes (2 and
3) being different!
Simplifying Radicals involving
Variables:
 Examples:
3
y 7 x5 z 6  3 y 3 y 3 yx3 x 2 z 3 z 3  yyxzz 3 yx 2  y 2 xz 2 3 yx 2
This is really what is taking place,
however, we usually don’t show
all of these steps! The easiest
thing to do is to divide the
exponents of the radicand by
the index. Any “whole parts”
come outside the radical.
“Remainder parts” stay
underneath the radical.
For instance, 3 goes into 7 two
whole times.. Thus y 2 will be
brought outside the radical.
There would be one factor of y
remaining that stays under the
radical.
Let’s get
some
more
practice!
Practice:
EX 1:
25 p 7  25  p 7  5 p 3 p
The index is 2. Square root of 25 is 5. Two goes into 7 three “whole” times, so a
p3 is brought OUTSIDE the radical.The remaining p1 is left underneath the radical.
EX 2:
4
32a5b7  4 25 a5b7  21 a1b1 4 2ab3  2ab 4 2ab3
The index is 4. Four goes into 5 one “whole” time, so a
2 and a are brought OUTSIDE the radical. The remaining 2 and a are left
underneath the radical. Four goes into 7 one “whole” time, so b is brought
outside the radical and the remaining b3 is left underneath the radical.
Simplifying Radicals by Using Smaller
Indexes:
 Sometimes we can
rewrite the expression
with a rational exponent
and “reduce” or simplify
using smaller numbers.
Then rewrite using
radicals with smaller
indexes:
12
2 2 2 4  4 2
3
3
12
1
More examples:
EX 1:
6
t t t  t
2
2
6
1
3
3
EX 2:
9
5  5  5 3  3 52  3 25
6
6
9
2
Multiplying Radicals with Difference
Indexes:
 Sometimes radicals can
be MADE to have the
same index by rewriting
1
2
1
3
3
2
2  6  2 6  2 6
first as rational
exponents and getting a
common denominator.
 6 22  6 63  6 4  216
Then, these rational
exponents may be
rewritten as radicals
with the same index in
order to be multiplied.
6 6
3
 6 864
Applications of Radicals:
 There are many applications of radicals.
However, one of the most widely used
applications is the use of the Pythagorean
Formula.
 You will also be using the Quadratic Formula
later in this course!
 Both of these formulas have radicals in them.
To learn more about them you may go to:
Pythagorean Theorem
What is the Pythagorean Formula?
Quadratic Formula