Post on 12-Nov-2014
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5.6 Simplifying Radicals
Definitions
A perfect square is the square of a natural number. 1, 4, 9, 16, 25, and 36 are the first six perfect squares.
A perfect cube is the cube of a natural number. 1, 8, 27, 64, 125, and 216 are the first six perfect cubes.
Perfect Powers
A quick way to determine if a radicand xn is a perfect power for an index is to determine if the exponent n is divisible by the index of the radical.
Example: 5 20x Since the exponent, 20, is divisible by the index, 5, x20 is a perfect fifth power.
This idea can be expanded to perfect powers of a variable for any radicand.The radicand xn is a perfect power when n is a multiple of the index of the radicand.
Product Rule for Radicals
Examples:
3333 424832
4444 3231648
3333 252125250
and numbers real enonnegativFor nnn abba
ba
,
Product Rule for Radicals
1. If the radicand contains a coefficient other than 1, write it as a product of the two numbers, one of which is the largest perfect power for the index.
2. Write each variable factor as a product of two factors, one of which is the largest perfect power of the variable for the index.
3. Use the product rule to write the radical expression as a product of radicals. Place all the perfect powers under the same radical.
4. Simplify the radical containing the perfect powers.
To Simplify Radicals Using the Product Rule
Product Rule for Radicals
Examples:
2623623672
4 354 34 204 3204 23 bbbbbbb
3233 633 63 222816 xyyxyx
4 324 28164 3118 21632 yxyxyx 4 3274 22 yxyx *When the radical is simplified, the
radicand does not have a variable with an exponent greater than or equal to the index.
Quotient Rule for Radicals
Examples:
10
9
100
81
100
81
0 ,
and numbers real enonnegativFor
bb
a
b
a
ba
nn
n
,
3
75 5251
25Simplify radicand, if possible.
Quotient Rule for Radicals
More Examples:
4
2
3 12
3 6
312
6 46464
y
x
y
x
y
x
2
42
4 8
44 8
4 8
4 8
48
8
4132
56
2
3
16
3
16
3
16
3
16
3
b
a
b
a
b
a
b
a
ba
ba
241
216
1
32
2
64
2
64 22
3
5
3
5
xxx
x
x
x
x
Adding, Subtracting, and Multiplying
Radicals
Like Radicals
444 292425
Like radicals are radicals having the same radicands. They are added the same way like terms are added.
Example:
2222 85103 xyzxyzxyzxyz 55 6776 Cannot be simplified further.
Adding & Subtracting
101651025316052503
Examples:
1. Simplify each radical expression.
2. Combine like radicals (if there are any).
To Add or Subtract Radicals
10351020101510451053
44 4444 44 454 5 331633483 xyxxxxyyxxxy
42444 3363323 xyxxxyxxyxxxy
)6(3 24 yxxyx
CAUTION!
The product rule does not apply to addition or subtraction!
baba
baba
Multiplying Radicals
)5(3 x
Multiply:
Use the FOIL method.
x353
)26)(58(
10562848
)63)(63( 36363632
33363 Notice that the inner and outer terms cancel.
Multiplying Radicals
More Examples:
2842 333
5 1638375 78135 93024 zyxzyxzyx
5 323775 325 153535 zyxzyxzyxzyx
39381273 232 yyyyyyyy )(
Dividing Radicals
Rationalizing Denominators
Examples:
To Rationalize a Denominator
3
6
3
3
3
2
3
2
233
32
3
3
3
2
3
2
y
yx
y
yxy
y
yx
y
y
y
x
y
x
Multiply both the numerator and the denominator of the fraction by a radical that will result in the radicand in the denominator becoming a perfect power.
r
prq
r
rpq
r
rpq
r
r
r
pq
r
pq
2
10
2
10
2
10
2
2
2
5
2
5 24444
Cannot be simplified further.
Conjugates
When the denominator of a rational expression is a When the denominator of a rational expression is a binomial that contains a radical, the denominator is binomial that contains a radical, the denominator is rationalized. This is done by using the rationalized. This is done by using the conjugate of of the denominator. The conjugate of a binomial is a the denominator. The conjugate of a binomial is a binomial having the same two terms with the sign binomial having the same two terms with the sign of the second term changed.of the second term changed.
The conjugate of 65 is 65
The conjugate of 44 23 is 23 yxyx
12
125
12
12
12
5
)(
Simplifying Radicals
Simplify by rationalizing the denominator:
12
5
dc
dc
2
dc
dcdcdc
dcdc
dcdc
22
))((
))(2(
dc
dc
dc
dc 2
Simplifying Radicals
A Radical Expression is Simplified When the Following Are All True
1. No perfect powers are factors of the radicand and all exponents in the radicand are less than the index.
2. No radicand contains a fraction.
3. No denominator contains a radical.
Simplifying Radicals
Simplify:
3
3
3
1
3
3
3
3
3
1
3
32
3
3
3
3
6
1004
3
82
6
6
6
1004
3
3
3
82
6
6004
3
242
6
61004
6
644
6
6104
6
624
6
640
6
683
616
6
632
3 5
6 5
3
3
)(
)(
r
r
35
65
3
3
)(
)(
r
r )()()( 35653r 6561065 33 )()( )()( rr
65)3(
1
r
Rational Exponents
Changing a Radical Expression
When a is nonnegative, n can be any index.When a is negative, n must be odd.
nn aa1
77 21
A radical expression can be written using exponents by using the following procedure:
3143 4 yxyx
9149 4 7373 zxzx
Changing a Radical Expression
When a is nonnegative, n can be any index.When a is negative, n must be odd.
nn aa1
15 15 21
Exponential expressions can be converted to radical expressions by reversing the procedure.
331 bb
Simplifying Radical Expressions
73 2372 9898 yxyx
This rule can be expanded so that radicals of the form can be written as exponential expressions. n ma
For any nonnegative number a, and integers m and n,
nmmnn m aaa
Power
Index
3 232 bb
Rules of Exponents
The rules of exponents from Section 5.1 also apply when the exponents are rational numbers.
For all real numbers a and b and all rational numbers m and n,
Product rule: am • an = am + n
Quotient rule:
Negative exponent rule:
0 , aaa
a nmn
m
0 ,1
aa
a mm
Rules of Exponents
For all real numbers a and b and all rational numbers m and n,
Zero exponent rule: a0 = 1, a 0
Raising a power to a power:
Raising a product to a power :
Raising a quotient to a power :
nmnm aa
mmm baab
0 ,
bb
a
b
am
mm
Rules of ExponentsExamples:
1.) Simplify x-1/2x-2/5.
1091091041055221-5221 1
xxxxxx
2.) Simplify (y-4/5)1/3.
154
15431543154 1
yyyy
3.) Multiply –3a-4/9(2a1/9 – a2). 914
31914932949194 3
63636 a
aaaaa
Factoring Expressions
Examples:
1.) Factor x1/4 – x5/4.
x1/4 – x5/4 = x1/4 (1 – x5/4-(1/4))
The smallest of the two exponents is 1/4.
Original exponent
Exponent factored out
= x1/4 (1 – x4/4) =
2.) Factor x-1/2 + x1/2.
x-1/2 + x1/2 = x -1/2 (1– x1/2-(-1/2) ) = 21
1
x
xx -1/2 (1– x) =
Original exponent
Exponent factored out
The smallest of the two exponents is -1/2.
x1/4 (1 – x)
Solving Radical Equations
Radical Equations
A radical equation is an equation that contains a variable in a radicand.
To solve radical equations such as these, both sides of the equation are squared.
9x 172 x 243 yy
229x
81x
22172 x
2892 x
291x
9x 172 x 243 yy
22243 yy
4443 2 yyy
yy 70 2 )7(0 yy 7 ; 0 yy
Extraneous Roots
In the previous example, an extraneous root was obtained when both sides were squared. An extraneous root is not a solution to the original equation. Always check all of your solutions into the original equation.
7 ; 0 yy
243 yyCheck:y = 0
20403 24
FALSE!
243 yyCheck:y = 7
27473 5421
525
Two Square Root Terms
To solve equations with two square root terms, rewrite the equation, if necessary so that there is only one term containing a square root on each side of the equation.
252 cc
22252 cc
Solve the equation:
252 cc
7c
Check:c = 7 275)7(2
9514
99
33
Nonradical Terms
1243 bb
Solve the equation:
Check:b = 84 22
1243 bb
12128163 bbb
12820 bb
22 12820 bb
)12(64400402 bbb
64128400402 bbb
0336882 bb0484 ))(( bb
4 ,84 bb
1243 bb
18424384 169481
1349
99 Not a solution.
b = 4 142434 941
341 11
Summary
To Solve Radical Equations
1. Rewrite the equation so that one radical containing a variable is isolated on one side of the equation.
2. Raise each side of the equation to a power equal to the index of the radical.
3. Combine like terms.
4. If the equation still contains a term with a variable in a radicand, repeat steps 1 through 3.
5. Solve the resulting equation for the variable.
6. Check all solutions in the original equation for extraneous solutions.