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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.

Examples:

3333 424832

4444 3231648

3333 252125250

and numbers real enonnegativFor nnn abba

ba

,

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

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.

Examples:

10

9

100

81

100

81

0 ,

and numbers real enonnegativFor

bb

a

b

a

ba

nn

n

,

3

75 5251

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

444 292425

Example:

2222 85103 xyzxyzxyzxyz 55 6776 Cannot be simplified further.

101651025316052503

Examples:

2. Combine like radicals (if there are any).

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

)5(3 x

Multiply:

Use the FOIL method.

x353

)26)(58(

10562848

)63)(63( 36363632

33363 Notice that the inner and outer terms cancel.

More Examples:

2842 333

5 1638375 78135 93024 zyxzyxzyx

5 323775 325 153535 zyxzyxzyxzyx

39381273 232 yyyyyyyy )(

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

)(

Simplify by rationalizing the denominator:

12

5

dc

dc

2

dc

dcdcdc

dcdc

dcdc

22

))((

))(2(

dc

dc

dc

dc 2

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.

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

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

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

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)

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

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