Download M098 Carson Elementary and Intermediate Algebra 3e Section 10.3 Objectives

M098
Carson Elementary and Intermediate Algebra 3e
Section 10.3
Objectives
1.
2.
3.
Multiply radical expressions.
Divide radical expressions.
Use the product rule to simplify radical expressions
Vocabulary
Prior Knowledge
First twenty perfect squares.
First seven perfect cubes.
New Concepts
1. Multiply radical expressions.
Product Rule: n a  n b  n a  b
Notice that the product rule only applies if the indexes are the same.
Example 1:
a.
3  12 
36  6
c.
8a  5b 
40ab
b.
5 7 
35
d. 3 xy  3 7x  3 7x 2 y
2. Divide radical expressions.
n
Quotient Rule:
n
a

n
b
a
b
Again notice that the index must be the same for the quotient rule to apply. The product rule and the
quotient rule can be used either way so we use whichever form is easiest for the given problem.
Example 2:
49
a.
64

7
8
b.
3

243

3

15
9

21

3
81
c.
243
15
81  9
d.
21
3
7
3. Use the product rule to simplify radical expressions.
The product rule provides the basis for simplifying radicals.
To simplify square roots, we look for the largest perfect square that divides into the number evenly. It’s
helpful if you learn at least the first 15 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169,
196, 225.
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M098
Carson Elementary and Intermediate Algebra 3e
Section 10.3
Always look for the largest perfect square that divides into the radicand evenly.
20 
45 
4  5  2 5
60 
4  15 
4  15  2 15
48 
16  3 
16  3  4 3
Although 4 is a perfect square that divides into 48 evenly, it is not the largest perfect square. Always look
for the largest perfect square that divides the number evenly.
245 
Sometimes (as in this case) it’s difficult to know which perfect square will divide into the number.
There is another way to do this. We know that the square root of 36 is 6 because 6 x 6 = 36. In other
words, for every two of the same factor under the radical, one can be brought to the outside. So if we find
the prime factors (numbers that only 1 and the number divide into evenly - 2, 3, 5, 7, 11, 13, 17, 19, etc.) of
the number, for every 2 of the same factor under the radical, one can be brought outside. An easy way to
find the prime factors is to create a “factor tower”. Basically, we do repeat division, dividing only by prime
numbers, until you get a prime number for an answer.
For example, to find 20 , create a tower like this to find the prime factorization of 20:
5
2 10
2
20
20 = 2 · 2 · 5
20 
225  2 5
Because there are 2 twos under the radical, the expression can be rewritten with one 2 outside the
radical.
Example 3:
60 
5
3 15
2 30
2
60 
60
2  2  3  5  2 3  5  2 15
Remember these are factors (multiplied) so if more than one number can be brought outside or if more
than one number is left under the radical, they are multiplied together.
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M098
Carson Elementary and Intermediate Algebra 3e
Section 10.3
Example 4:
48 
22223  22 3  4 3
So back to the earlier problem:
245 
577  7 5
It’s faster to use the product rule, but if you don’t see the largest perfect square factor, the factor tower will
always work. Here are some examples of simplifying radicals doing it using both methods.
300 
100 3  10 3
300 
2  2  3  5  5  2  5 3  10 3
52 
4 13  2 13
52 
2  2  13  2 13
108 
36 3  6 3
108 
22333  23 3  6 3
The same process works for roots with higher indexes. For cube (3rd) roots, we need three of the same
factors under the radical to bring one to the outside. To use the product rule, we need to know the perfect
cubes: 1, 8, 27, 64, 125…
3
3
250 
3
250 
3
125
3
2 5
3
2
5552  5
3
2
For higher radicals, don’t forget to write the index in the answer.
3
3
243 
3
3
3
243  3 3  3  3  3  3  3 3  3  3 9
27
9 3
3
3
9
We can do the same thing with variables. Every pair of identical factors makes a square.
x3 
xxx  x x
x5 
x  x x  x x  x  x x  x2
x
When we’re working with square roots, we’re asking “How many groups of 2 are there?”. That’s just a
division problem, so mentally we think 5 / 2 = 2 with 1 left over. That means we can bring 2 x’s outside
with one left inside.
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M098
Carson Elementary and Intermediate Algebra 3e
Section 10.3
x9 
Think9  2  4 R 1
That means 4 to the outside and 1 left inside
x9 
x  x  x  x  x  x  x  x  x  x  x  x  x x  x4
x
When we combine numbers and variables, do the numbers first and then the variables.
75x 7 y 3 
x7
25  3
 5 3 x3 x
y3
y y
 5x 3 y 3xy
It is helpful when simplifying radicals to write the radical for the answer first including the index if it is a
number other than 2, leaving some space in front of the radical, because some factors will be pulled out of
the radical and some will remain inside the radical.
48a 3b 7 c 4
Example 5: Simplify
48a 3b 7 c 4 
48 = 16  3. Since the square root of 16 is 4, the 4 comes outside and the 3 stays inside.
 4
3
There is one pair of a’s, so one a is written outside and one stays on the inside.
 4a
3a
There are three pairs of b’s, so three b’s are written outside and one stays on the inside.
 4ab3
3ab
There are two pairs of c’s, so two c’s are written outside and there are none left inside.
 4ab3 c 2 3ab
Example 6: Simplify
3
24x 2 y 7 z 5
This is a cube root so the factors of 24 to use are 8 and 3. Since the cube root of 8 is 2,
the 2 is written outside the radical.
2
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M098
Carson Elementary and Intermediate Algebra 3e
Section 10.3
There aren’t enough x’s to bring any to the outside.
3
2
3x 2
There are two groups of three y’s with one y left over so two y’s are written outside and
one y stays inside the radical.
3
 2y 2
3x 2 y
There is one group of three z’s so one z is written outside the radical and two are left
inside.
 2y 2 z
3
3x 2 yz 2
Find the product and write the answer in simplest form.
Example 7:
6 15c 2  2 10c 5
12 150c 7
Multiply the coefficients and the radicands.
12 25  6  c 6  c
Rewrite the radicand with perfect square factors.
12  5  c 3 6  c
Simplify the perfect square factors.
60c 3 6c
Multiply factors.
Example 8:
54 240r 9 s10
9 5r 6 s 4
54
9
240r 9 s10
5r 6 s 4
Divide the coefficients and the radicands.
6 48r 3 s 6
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6 16  3  r 2  r  s 6
Rewrite the radicand with perfect square factors.
6  4  r  s3 3r
Simplify the perfect square factors.
24rs 3 3r
Multiply factors.
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