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Chapter 10
Radicals and Rational Radicals and Rational ExponentsExponents
10.1 Finding Roots
10.2 Rational Exponents
10.3 Simplifying Expressions Containing Square Roots
10.4 Simplifying Expressions Containing Higher Roots
10.5 Adding, Subtracting, and Multiplying Radicals
10.6 Dividing RadicalsPutting it All Together
10.7 Solving Radical Equations
10.8 Complex Numbers
1010 Radicals and Rational Radicals and Rational ExponentsExponents
Simplifying Expressions Containing Square RootsSimplifying Expressions Containing Square Roots10.310.3
Multiply Square Roots
.164product thebegin with sLet'
There are two ways that the product can be found. Let’s take a look at both ways.
First method is to find the square roots of each and then multiply the results.
164 42 8
Second method is to multiply the radicands and then find the square root.
164 164 864 Notice that both will obtain the same results. This leads us to the product rule forMultiplying expressions containing square roots.
Example 1
mba 5)37)
Multiply.
Solution
37)a 37 21
mb 5) m5 m5
BeCareful
We can multiply radicals this way only if the indices are the same. Wewill see later how to multiply radicals with different indices such as 321 t
Simplify the Square Root of a Whole Number
How do we know when a square root is simplified?
To simplify expressions containing square roots, we reverse the process of multiplying. That is we use the product rule that says where a andb is a perfect square.
baba
Example 2
15)500)75)45) dcba
Simplify completely.
Solution
a) The radical is not in simplest form since 45 contains a factor (other than 1) That is a perfect square. Think of two numbers that multiply to 45 so that at leastOne of the numbers is a perfect square:
45
5945
Notice, however that ,but neither 3 or 15 is a perfect square. So we need to use 9 and 5.
15345
45 59
59
53
9 is a perfect square.
Product Rule.
39
squares.perfect
are that factorsany havenot does 5 because simplified completely is 53
Examples b through d continued on next page…
Example 2
15)500)75)45) dcba
Simplify completely.
Solution
b) The radical is not in simplest form since 75 contains a factor (other than 1) Think of two numbers that multiply to 75 so that at least one of the numbers is a perfect square:
75
32575
Notice, 25 is a perfect square and 3 is not a perfect square.
75 325
325
35
25 is a perfect square.
Product Rule.
525
squares.perfect
are that factorsany havenot does 5 because simplified completely is 35
Examples c continued on next page…
Example 2
15)500)75)45) dcba
Simplify completely.
Solution
c) The radical is not in simplest form since 500 contains a factor (other than 1). Think of two numbers that multiply to 500 so that at least one of the numbers is perfect square:
500
5100500
Notice, 100 is a perfect square and 5 is not a perfect square.
500 5100
5100
510
25 is a perfect square.
Product Rule.
10100
squares.perfect
are that factorsany havenot does 5 because simplified completely is 510
Examples c continued on next page…
d) Does 48 have a factor that is a perfect square? 31648 Yes,
asit rewrite ,48simpify To
48 316
316 34
16 is a perfect square.
Product Rule.
416
34However, notice that .12448
48 124124
122122
Notice that is not in simplest form. We must continue to simplify.
12
48 122342
342 322
34
Example 2-Continued
48)15)75)45) dcbaSimplify completely.
Solution
Use the Quotient Rule for Square Roots
roots. square containing sexpression dividingfor rulequotient the tous leads This
.249
36
9
36 that truealso isIt .2
3
6
9
36say can We.
9
36simplify sLet'
Example 3
49
9)a
Simplify completely.
Solution
49
9
49
9
7
3
Quotient Rule
Since 9 and 49 are perfect squares, find the square root of each separately.
749 and 39
Example 4
3
300)a
Simplify completely.
Solution
3
300100
10
Simplify
10
120)b
36
5)c
a) Neither 300 nor 3 are perfect squares, so you want to simplify to get 100, which is a perfect square. 3
300
3
300
b) Neither 120 nor 10 are perfect squares. There are two methods that you can use to simplify One is to apply the quotient rule to obtain a fraction under one radical and then simplify and the second is to apply the product rule to rewrite each radical and then simplify the fraction.
10
120
10
120
12 34 32
Quotient rule
Square root of 4 is 2.
10
120
25
304
25
652
25
2352
32
Product rule
Divide out common Factors.
Method 2Method 1
Example 4-Continued
3
300)a
Simplify completely.
10
120)b
36
5)c
Solution
c) The fraction does not reduce. However, 36 is a perfect square. Begin by applying the quotient rule.
36
5
36
5
36
5
6
5
Quotient rule
636
Multiplying is the same as dividing 6 by 2. We can generalize this result withThe following statement.
Simplify Square Root Expressions Containing Variables with Even Exponents
A square root is not simplified if it contains any factors that are perfect squares.This means that a square rot containing variables is simplified if the power on eachvariable is less than 2. For example, is not in simplified form. If r represents anonnegative real number, then we can use rational exponents to simplify .
2/16
6r6r
6r 32/62/162/16 rrrr
We can combine this property with the product and quotient rules to simplify radicalexpressions.
Example 5
2) ba
Simplify completely.
Solution
4100) ab 824) pc
2) ba 2
12
b 2/2b 1b b
4100) ab 4100 a 2/410 a 210a
824) pc 824 p
2/864 p
462 p
62 4p
6
27)g
d
6
27)g
d6
27
g
2/6
39
g
2/6
39
g
3
33
g
Product rule
4 is a perfect square.
Simplify.
Rewrite using the commutative property.
Begin by using the quotient rule.
Simplify Square Root Expressions Containing Variables with Odd Exponents
How do we simplify an expression containing a square root if the power under theSquare root is odd? We can use this product rule for radicals and fractional exponents to help us understand how to simplify such expressions.
Example 6
9) ba
Simplify completely.
13) ab
9) ba
428
SolutionTo simplify the radicals , write the variable as the product of two factors so that the exponent of one of the factors is the largest numbers less than 9 that is divisible by 2 (the index of the radical).
18 bb
bb 8
bb 2/8
bb4
8 is the largest number lessthan 9 that is divisible by 2.
Product Rule
Use a fractional exponentto simplify.
13) ab
6212
112 aa
aa 12
aa 2/12
aa6
12 is the largest number less than 9 that is divisible by 2.
Product Rule
Use a fractional exponent to simplify.
If you notice in the previous examples, we always divided by 2. Let’s look at theprevious examples to see if we can used division to help us simplify radicals.
9b 18 bb bb4
1
8
492
Quotient
Remainder
Index of radical
13a 112 aa aa6
1
12
6132
Quotient
Remainder
Index of radical
Example 7
5) ka
Simplify completely.
349) vb 1932) mcSolution
:divide ,simplify To) 5ka25241kkkkk 2125
349) vb 349 v
vv7
vv7
Product Rule
1. ofremainder a and 2 ofquotient a gives 23
1932) mc 1932 m
mm9216
mm924
Product Rule
1. ofremainder a and 9 ofquotient a gives 219
mm 24 9
mm 24 9
416
Use commutative property to rewrite expression
Use the product rule to write the expression with one radical.
Example 8
9718) nma
Simplify completely.
12
93)
g
fb
9718) nma
Solution
9718 nm 141329 nnmm
Use the Product Rule for each Radical.
141323 nnmm 39
nmnm 23 43
mnnm 23 43
Use commutative property to rewrite expression
Use the product rule to write the expression with one radical.
12
93)
g
fb
12
93
g
f
12
93
g
f
6
143
g
ff
6
14 3
g
ff
6
4 3
g
ff
Quotient Rule
Product Rule
Use commutative property to rewrite expression
Use the product rule to write the expression with one radical.
Simplify More Square Root Expressions Containing VariablesExample 9
bba 28)
Simplify completely.
45 63) xyyxb 3
7
2
36)
r
rc
Solutionbba 28) bb 28
216b
216 b
b4
b4
Product Rule
Product Rule
45 63) xyyxb 5618 yx5618 yx
5629 yx 5629 yx
yyx 2323 yyx 23 23 yyx 23 23
Multiply the radicands together to obtain one radical.
Product Rule
Product Rule
Evaluate Commutative property
Example 9-Continued
bba 28)
Simplify completely.
45 63) xyyxb 3
7
2
36)
r
rc
Solution
3
7
2
36)
r
rc
3
7
2
36
r
r
418r
429 r
223 r
23 2r
Use Quotient Rule.
Product Rule.
Commutative property