Transcript Ch. 5.6: Radical Expressions

Ch. 5.6: Radical Expressions
Product Property
n
ab  a
n
n
b
**When multiplying two radicals, multiply the
coefficients, then multiply the radicals**
n
Quotient Property
n
a
a
 n
b
b
**If there is a radical in the denominator, you
must multiply both the numerator and
denominator by a radical that will get the
radical out of the denominator**
Simplifying non-perfect roots:
Numbers
• Tree factor the number to its prime factors
• See if you can make groups the size of the index
• One group makes one of the number outside,
anything not grouped goes back inside
1) 27 1. Tree factor
2) 3 108
3 3 3
3 3
2. Make groups the size of
The index
3.Group makes one outside,
Non-group items stays inside
9  12
3
3 3 4  3
3
3 3 3 2  2
3
3
3 4
Simplifying non-perfect roots:
Variables
• Divide the variable exponent by the index
• Turn that number into a mixed number
Exponent
outside
1)
x
x
Exponent inside
Index
13
3
17 18
2) x y z
7
7
1
3
2
2
3
b
a
c
x
1. Divide exponent
by index
2. Follow picture
above
13 1
=4
3
3
17
2
5
3
3
4
1
5 63
x y z
4
2 0
xy z
5 63
x y z
18
0
6
3
3
xy
2
Simplifying non-perfect roots:
Numbers and Variables
• Split the numbers and variables into two different
radicals
• Solve each individually
• Write final answer with only one radical
13
1. Split into two problems
40 y
40
y13
4 10
2 25 2
2. Simplify each radical
13
1
6
2
2
y 6 y1
2 10
6
2 y 10 y
3. Combine into one radical
Simplifying Quotients
1. Simplify the radicals individually in the numerator and
denominator
2. If there is a radical in the denominator, follow these steps to
remove it
a. Tree factor numbers, divide variable exponents by the index
b. Ask yourself, how many more do I need to make a group for
the numbers, how many more do I need to make a group for
the variables
c. Place your answers in a radical, and multiply the numerator
and denominator by your answers
d. Simplify!
12
8
y
Ex : 3
6x
3
8y
12
4
2y
3
3
6x
6x
2 y4
3
6x
3
66xx
3
66xx
2 y 4 3 36 x 2
3
666xxx
1. Simplify individually
2. Denominator with radical!!!
a) How many 6’s are needed to
make a group of 3?
b) How many x’s are needed to
make a group of 3?
3. Multiply numerator and
Denominator by 3 6 6 x x
4. Simplify!
2 y 4 3 36 x 2
y 4 3 36 x 2

6x
3x
Adding/Subtracting Radicals
1) Simplify each radical
2) Combine radicals that are the same by
adding/subtracting the coefficients and
keeping the radical
Ex: 3 5  2 7  8 5 Ex: 20  12  45  48
11 5  2 7
2 2 5 2 2 3 3 3 5 4 4 3
2 5 2 3 3 5  4 3
 52 3
FOIL: Just like with variables
1)Numbers outside radical multiply together
2) Numbers inside radical multiply together
1.(2 6  3 2)(3  2)
2 6 3 2 6
2 3 2 3 3 2
2
6 6  2 12  9 2  3 4
1. Foil
6 6 2 4
2. Multiply using rules
above
3 9 2 3 2
6 6  2 2 3 9 2 3 2
3. Simplify the radicals