A5 - Rational Exponents VJ

8/11/2019 A5 - Rational Exponents VJ

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Radical Expressions and Rational Exponents

Mathematics 17

Institute of Mathematics, University of the Philippines-Diliman

Lecture 5

Math 17 (UP-IMath) Radical Expressions and Rational Exponents Lec 5 1 / 23

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Outline

1 Radical Expressionsnth root of a

Simplification of Radical ExpressionsOperations Involving Radical Expressions

Addition and Subtraction of Radical Expressions

Multiplication and Division of Radical Expressions

Rationalizing Radical Expressions


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nth root of a

Definition

Let a, b ∈ R and n ∈ N, n > 1. If bn = a then b is an nth root of a.


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nth root of a

Definition

Let a, b ∈ R and n ∈ N, n > 1. If bn = a then b is an nth root of a.

Examples:Since both 32, (−3)2 are equal to 9, then 3 and -3 are second roots orsquare roots of 9.

-4 is a third root or a cube root of -64 since (−4)3 = −64.

2 and -2 are fourth roots of 16 since (−2)4

= (2)4

= 16.


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Principal nth root of a

Definition

If n ∈ N, n > 1, a ∈ R, the principal nth root of a, denoted n

√ a, is defined

as follows:

If a > 0, then n

√ a is the nth root of a that is positive.

If a < 0 and n is odd, then n√ a is the nth root of a that is negative.

If a is zero, then n

√ 0 = 0.


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Definition

If n ∈ N, n > 1, a ∈ R, the principal nth root of a, denoted n

√ a, is defined

as follows:

If a > 0, then n

√ a is the nth root of a that is positive.

If a < 0 and n is odd, then n√ a is the nth root of a that is negative.

If a is zero, then n

√ 0 = 0.

Note:

1. The principal nth root is unique.2. If n is even, the nth roots of negative real numbers are not real

numbers.


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Radical Expression


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Radical Expression


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Radical Expression


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Radical Expression

If no index is written, it is taken to be 2.


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Examples:

4 and −4 are square roots of 16, but the principal square root of 16 is4.


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Examples:

4 and −4 are square roots of 16, but the principal square root of 16 is4. That is,

√ 16 = 4.


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Examples:


√ 16 = 4.

3 is the principal cube root of 27 since (3)3 = 27 and 3 is a positiveinteger. That is, 3√ 27 = 3.


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Examples:


√ 16 = 4.

3 is the principal cube root of 27 since (3)3 = 27 and 3 is a positiveinteger. That is, 3√ 27 = 3.3√ −27


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Examples:


√ 16 = 4.

3 is the principal cube root of 27 since (3)3 = 27 and 3 is a positiveinteger. That is, 3√ 27 = 3.3√ −27 = −3


P i i l h f

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Examples:


√ 16 = 4.

3 is the principal cube root of 27 since (3)3 = 27 and 3 is a positiveinteger. That is, 3√ 27 = 3.3√ −27 = −32011√

0


P i i l h f

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Examples:


√ 16 = 4.


0 = 0


P i i l h f

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Examples:


√ 16 = 4.


0 = 0√ −

9


P i i l th t f

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Examples:


√ 16 = 4.


0 = 0√ −

9 is not a real number because there is no real number b suchthat b2 = −9.


R ti l E t f R di l E i

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Rational Exponents for Radical Expressions

Definition

If n is a positive integer and m is an integer such that mn is in lowestterms, then

a1

n = n

√ a, and

am

n = a1

nm

= ( n√

a)m



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Definition

If n is a positive integer and m is an integer such that mn is in lowestterms, then

a1

n = n

√ a, and

am

n = a1

nm

= ( n√

a)m

Examples:

1. 251/2 =√

25 = 5

2. 272/3 = ( 3√

27)2 = 32 = 9

3. (−32)1/5 = 5√ −32 = −2

4. −(32)1/5 = − 5√

32 = −2

5. 4−3/2 = (√

4)−3 = 2−3 = 1

8



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The laws of integral exponents can be extended to rational exponents:

Laws of Rational Exponents

Let a, b ∈ R, r, s ∈ Q.

ar · as = ar+s

ar

as = ar−s, a = 0

a−r = 1

ar, a = 0

(ar)s = ars, a ≥ 0

(ab)r = arbra

b

r=

ar

br , b = 0


Radical Expression and the Rational Exponents

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Examples:

1. (x1/2 + y

1/2)(x1/2 − y

1/2)



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Examples:

1. (x1/2 + y

1/2)(x1/2 − y

1/2)

= (x1/2)2 − (y

1/2)2



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Examples:

1. (x1/2 + y

1/2)(x1/2 − y

1/2)

= (x1/2)2 − (y

1/2)2

= x − y



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Examples:1. (x

1/2 + y1/2)(x

1/2 − y1/2)

= (x1/2)2 − (y

1/2)2

= x − y

2. a + a1

/2 − a

1

/3

a2/3



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Examples:1. (x

1/2 + y1/2)(x

1/2 − y1/2)

= (x1/2)2 − (y

1/2)2

= x − y

2. a + a1

/2 − a

1

/3

a2/3

= a

a2/3+

a1/2

a2/3− a

1/3

a2/3



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Examples:1. (x

1/2 + y1/2)(x

1/2 − y1/2)

= (x1/2)2 − (y

1/2)2

= x − y

2. a + a1

/2 − a

1

/3

a2/3

= a

a2/3+

a1/2

a2/3− a

1/3

a2/3

= a1/3



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Examples:1. (x

1/2 + y1/2)(x

1/2 − y1/2)

= (x1/2)2 − (y

1/2)2

= x − y

2. a + a1

/2 − a

1

/3

a2/3

= a

a2/3+

a1/2

a2/3− a

1/3

a2/3

= a1/3 + a−

1/6



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Examples:1. (x

1/2 + y1/2)(x

1/2 − y1/2)

= (x1/2)2 − (y

1/2)2

= x − y

2. a + a1

/2 − a

1

/3

a2/3

= a

a2/3+

a1/2

a2/3− a

1/3

a2/3

= a1/3 + a−

1/6

−a−

1/3



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p p

Examples:1. (x

1/2 + y1/2)(x

1/2 − y1/2)

= (x1/2)2 − (y

1/2)2

= x − y

2. a + a1

/2

− a1

/3

a2/3

= a

a2/3+

a1/2

a2/3− a

1/3

a2/3

= a1/3 + a−

1/6

−a−

1/3

= a1/3 + 1a1/6

− 1a1/3


Simplification of Radical Expressions

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p p



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p p

We use the following properties:

Theorem

Let m, n ∈ N, n > 1 for any a, b ∈ R with a, b ≥ 0 if n is even.n

√ an = a;

n

√ ank = ak, k

∈Z

n√ ab = n√ a · n√ bn

a

b =

n

√ a

n

√ b

, b = 0

n

√ am = a

m/n

n

m√ a = mn√ a



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p p

Examples:1.√

2 · √ 18 =√

2 · 18 =√

36 = 6

2.3√

543√

2= 3

54

2 = 3

√ 27 = 3

3.

4√

x

3

= x

3/4

(x ≥ 0 if the index is even.)



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Examples:1.√

2 · √ 18 =√

2 · 18 =√

36 = 6

2.3√

543√

2= 3

54

2 = 3

√ 27 = 3

3.

4√

x3

= x

3/4

(x≥ 0 if the index is even.)

Note: if n is even, am/n = n

√ am is true only if a ≥ 0. If a < 0, the

statement is not necessarily true.

Example: 43/2 = (√ 4)3 = 23 = 8 and √ 43 = √ 64 = 8



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Examples:

1.√

2 · √ 18 =√

2 · 18 =√

36 = 6

2.3√

543√

2= 3

54

2 = 3

√ 27 = 3

3.

4√

x3

= x

3/4

(x≥ 0 if the index is even.)

Note: if n is even, am/n = n

√ am is true only if a ≥ 0. If a < 0, the

statement is not necessarily true.

Example: 43/2 = (√ 4)3 = 23 = 8 and √ 43 = √ 64 = 8

(−1)6/2 = (−1)3 = −1 but

(−1)6 =√

1 = 1



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A radical expression is in simplest form if all the following conditionsare satisfied:

The radicand has no factors which are perfect powers of the index n.

The radicand is positive for radicals with odd indices.

The radicand is not a fraction. (Rationalize the denominator.)

The smallest possible index is used.


Simplification of Radicals

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Example: 18x3y2



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Example: 18x3y2

=

32 · 2 · x2 · x · y2



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Example: 18x3y2

=

32 · 2 · x2 · x · y2

=√

32 · √ 2 ·√

x2 · √ x ·

y2



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Example: 18x3y2

=

32 · 2 · x2 · x · y2

=√

32 · √ 2 ·√

x2 · √ x ·

y2

= 3xy√

2x



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Example: 3 −8x7y2



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Example: 3 −8x7y2

= 3

(−2)3 · x6 · x · y2



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Example: 3 −8x7y2

= 3

(−2)3 · x6 · x · y2

= 3

(−2)3 · 3√

x6 · 3√ x · 3

y2



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Example: 3 −8x7y2

= 3

(−2)3 · x6 · x · y2

= 3

(−2)3 · 3√

x6 · 3√ x · 3

y2

= −2x2 3

xy2



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Example: 3

10y4

3x2



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Example: 3

10y4

3x2

= 3

10y4

3x2 · 3

2x

32x



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Example: 3

10y4

3x2

= 3

10y4

3x2 · 3

2x

32x =

3

90xy4

33 · x3



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Example: 3

10y4

3x2

= 3

10y4

3x2 · 3

2x

32x =

3

90xy4

33 · x3 =

3

90xy · y3

3√

33 · x3



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Example: 3

10y4

3x2

= 3

10y4

3x2 · 3

2x

32x =

3

90xy4

33 · x3 =

3

90xy · y3

3√

33 · x3

= y 3√

90xy

3x


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Example: 4√

9


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Example: 4√

9 = 4√

32


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Example: 4√

9 = 4√

32 = 32/4


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Example: 4√

9 = 4√

32 = 32/4 = 31/2


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Example: 4√

9 = 4√

32 = 32/4 = 31/2 =√

3


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Example: 4√

9 = 4√

32 = 32/4 = 31/2 =√

3

Example: 6√ 16x4


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Example: 4√

9 = 4√

32 = 32/4 = 31/2 =√

3

Example: 6√ 16x4 = 6

(2x)4


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Example: 4√

9 = 4√

32 = 32/4 = 31/2 =√

3


(2x)4 = 3

(2x)2


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Example: 4√

9 = 4√

32 = 32/4 = 31/2 =√

3


(2x)4 = 3

(2x)2 = 3√ 4x2



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Radicals with the same index and radicand can be added or subtracted.



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Examples:

1. 4√

3 − 5√

12 + 2√

75



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Examples:

1. 4√

3 − 5√

12 + 2√

75= 4

√ 3−

5(2√

3) + 2(5√

3)



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Examples:

1. 4√

3 − 5√

12 + 2√

75= 4

√ 3−

5(2√

3) + 2(5√

3)

= 4√ 3 − 10√ 3 + 10√ 3



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Examples:

1. 4√

3 − 5√

12 + 2√

75= 4

√ 3−

5(2√

3) + 2(5√

3)

= 4√ 3 − 10√ 3 + 10√ 3= 4

√ 3



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Examples:

1. 4√

3 − 5√

12 + 2√

75= 4

√ 3−

5(2√

3) + 2(5√

3)

= 4√ 3 − 10√ 3 + 10√ 3= 4

√ 3

2. 5√

x3 −√

121x3 +√

16x



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Examples:

1. 4√

3 − 5√

12 + 2√

75= 4

√ 3−

5(2√

3) + 2(5√

3)

= 4√ 3 − 10√ 3 + 10√ 3= 4

√ 3

2. 5√

x3 −√

121x3 +√

16x= 5x

√ x

−11x

√ x + 4

√ x



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Examples:

1. 4√

3 − 5√

12 + 2√

75= 4

√ 3−

5(2√

3) + 2(5√

3)

= 4√ 3 − 10√ 3 + 10√ 3= 4

√ 3

2. 5√

x3 −√

121x3 +√

16x= 5x

√ x

−11x

√ x + 4

√ x

= −6x√ x + 4√ x



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If radicals have the same index, we use

n

√ a· n

√ b

=

n

√ ab

and

n

√ a

n√ b =

n ab

Examples:

1. 3

3x2y · 3√ 36x



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n

√ a· n

√ b

=

n

√ ab

and

n

√ a

n√ b =

n ab

Examples:

1. 3

3x2y · 3√ 36x

= 3 3x2y·

36x


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n

√ a· n

√ b

=

n

√ ab

and

n

√ a

n√ b =

n ab

Examples:

1. 3

3x2y · 3√ 36x

= 3 3x2y·

36x = 3 33 · 22

·x3

·y

= 3x 3√ 4y



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If radicals have the same index, we use n

√ a· n

√ b = n

√ ab and

n

√ a

n√ b= n a

b

Examples:

1. 3

3x2y · 3√ 36x

= 3 3x2y·

36x = 3 33 · 22

·x3

·y

= 3x 3√ 4y

2.

√ 15x · √ 2x√

6x3



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√ a· n

√ b = n

√ ab and

n

√ a

n√ b= n a

b

Examples:

1. 3

3x2y · 3√ 36x

= 3 3x2y·

36x = 3 33 · 22

·x3

·y

= 3x 3√ 4y

2.

√ 15x · √ 2x√

6x3

=

√ 15x

·2x

√ 6x3



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√ a· n

√ b = n

√ ab and

n

√ a

n√ b= n a

b

Examples:

1. 3

3x2y · 3√ 36x

= 3 3x2y

·36x = 3 33 · 22

·x3

·y

= 3x 3√ 4y

2.

√ 15x · √ 2x√

6x3

=

√ 15x

·2x

√ 6x3 =

√ 30x2

√ 6x3



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√ a· n

√ b = n

√ ab and

n

√ a

n√ b= n a

b

Examples:

1. 3

3x2y · 3√ 36x

= 3 3x2y

·36x = 3 33 · 22

·x3

·y

= 3x 3√ 4y

2.

√ 15x · √ 2x√

6x3

=

√ 15x

·2x

√ 6x3 =

√ 30x2

√ 6x3 = 30x2

6x3



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√ a· n

√ b = n

√ ab and

n

√ a

n√ b= n a

b

Examples:

1. 3

3x2y · 3√ 36x

= 3 3x2y

·36x = 3 33 · 22

·x3

·y

= 3x 3√ 4y

2.

√ 15x · √ 2x√

6x3

=

√ 15x

·2x

√ 6x3 =

√ 30x2

√ 6x3 = 30x2

6x3 = 5

x · x

x



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√ a· n

√ b = n

√ ab and

n

√ a

n√ b= n a

b

Examples:

1. 3

3x2y · 3√ 36x

= 3 3x2y

·36x = 3 33 · 22

·x3

·y

= 3x 3√ 4y

2.

√ 15x · √ 2x√

6x3

=

√ 15x

·2x

√ 6x3 =

√ 30x2

√ 6x3 = 30x2

6x3 = 5

x · x

x

=

√ 5x

x



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If radicals have different indices, we first make their indices the same byfinding the LCM of all the indices, then using n

√ a = nm

√ am.

Example:√

2 · 4√ 8



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√ a = nm

√ am.

Example:√

2 · 4√ 8LCM of indices: 4



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√ a = nm

√ am.

Example:√


√ 2 · 4√ 8 = 4√ 22 · 4√ 8



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√ a = nm

√ am.

Example:√


√ 2 · 4√ 8 = 4

√ 22 · 4√ 8= 4

4(8)



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√ a = nm

√ am.

Example:√


√ 2 · 4√ 8 = 4

√ 22 · 4√ 8= 4

4(8)

= 4√

32



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√ a = nm

√ am.

Example:√


√ 2 · 4√ 8 =

4

√ 22 · 4√ 8

= 4

4(8)

= 4√

32

= 4√

25



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√ a = nm

√ am.

Example:√


√ 2 · 4√ 8 =

4

√ 22 · 4√ 8

= 4

4(8)

= 4√

32

= 4√

25

= 2 4√

2


Rationalizing a Denominator

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If denominator consists of one radicalMultiply both the numerator and denominator by an expression that willmake the radicand of the denominator a perfect power of the index.

Example: 16x4

3z



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Example: 16x4

3z

= 4x2

√ 3z



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Example: 16x4

3z

= 4x2

√ 3z

·√

3z√ 3z



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Example: 16x4

3z

= 4x2

√ 3z

·√

3z√ 3z

= 4x2

√ 3z√

32z2



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Example: 16x4

3z

= 4x2

√ 3z

·√

3z√ 3z

= 4x2

√ 3z√

32z2

=

4x2√

3z

3z

Math 17 (UP IMath) Radical Expressions and Rational Exponents Lec 5 20 / 23


If denominator consists of two or more radicals

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To rationalize the denominator, use the special products:(x − y)(x + y) = x2 − y2

(x ± y)(x2 ∓ xy + y2) = x3 ± y3




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(x ± y)(x2 ∓ xy + y2) = x3 ± y3

Example:

√ 3x +

√ 2x

√ 3x −√ 2x

=

√ 3x +

√ 2x√

3x −√ 2x




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(x ± y)(x2 ∓ xy + y2) = x3 ± y3

Example:

√ 3x +

√ 2x

√ 3x −√ 2x

=

√ 3x +

√ 2x√

3x −√ 2x

·




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(x ± y)(x2 ∓ xy + y2) = x3 ± y3

Example:

√ 3x +

√ 2x

√ 3x −√ 2x

=

√ 3x +

√ 2x√

3x −√ 2x

·√

3x +√

2x√ 3x +

√ 2x




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(x ± y)(x2 ∓ xy + y2) = x3 ± y3

Example:

√ 3x +

√ 2x

√ 3x −√ 2x

=

√ 3x +

√ 2x√

3x −√ 2x

·√

3x +√

2x√ 3x +

√ 2x

= (√

3x +√

2x)2

3x − 2x




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(x ± y)(x2 ∓ xy + y2) = x3 ± y3

Example:

√ 3x +

√ 2x

√ 3x −√ 2x

=

√ 3x +

√ 2x√

3x −√ 2x

·√

3x +√

2x√ 3x +

√ 2x

= (√

3x +√

2x)2

3x − 2x

=

3x + 2√

3x

·2x + 2x

x




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(x ± y)(x2 ∓ xy + y2) = x3 ± y3

Example:

√ 3x +

√ 2x

√ 3x −√ 2x

=

√ 3x +

√ 2x√

3x −√ 2x

·√

3x +√

2x√ 3x +

√ 2x

= (√

3x +√

2x)2

3x − 2x

=

3x + 2√

3x

·2x + 2x

x =

5x + 2x√

6

x

M th 17 (UP IM th) R di l E ssi s d R ti l E ts L 5 21 / 23



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(x ± y)(x2 ∓ xy + y2) = x3 ± y3

Example:

√ 3x +

√ 2x

√ 3x −√ 2x

=

√ 3x +

√ 2x√

3x −√ 2x

·√

3x +√

2x√ 3x +

√ 2x

= (√

3x +√

2x)2

3x − 2x

=

3x + 2√

3x

·2x + 2x

x =

5x + 2x√

6

x

= 5 + 2√

6

M th 17 (UP IM th) R di l E i d R ti l E t L 5 21 / 23


Example: 1

2 − 3√

4

http://find/



= 1

2 − 3√ 4



Example: 1

2 − 3√

41 4 3

√ 4

3√

42

http://find/



= 1

2 − 3√ 4 · 4 + 2 3

√4 +

3√

42

4 + 2 3√ 4 + 3√ 42



Example: 1

2 − 3√

41 4 2 3

√ 4

3√

42

http://find/



= 1

2 − 3√ 4 · 4 + 2 3

√4 +

3√

42

4 + 2 3√ 4 + 3√ 42

= 4 + 2 3

√ 4 +

3√

42

8 − 4

M h 17 (UP IM h) R di l E i d R i l E L 5 22 / 23


Example: 1

2 − 3√

41 4 + 2 3

√ 4 +

3√

42

http://find/



= 1

2 − 3√ 4 · 4 + 2 3

√4 +

3√

42

4 + 2 3√ 4 + 3√ 42

= 4 + 2 3

√ 4 +

3√

42

8 − 4

= 4 + 2 3√ 4 +

3√ 42

4

M h 17 (UP IM h) R di l E i d R i l E L 5 22 / 23


Example: 1

2 − 3√

41 4 + 2 3

√ 4 +

3√

42

http://find/



= 1

2 − 3√ 4 · 4 + 2 3

√4 +

√42

4 + 2 3√ 4 + 3√ 42

= 4 + 2 3

√ 4 +

3√

42

8 − 4

= 4 + 2 3√ 4 +

3√ 424

= 1 +3√

4

2 +

3√

16

4



Example: 1

2 − 3√

41 4 + 2 3

√ 4 +

3√

42

http://find/



= 1

2 − 3√ 4 · 4 + 2

√4 +

√42

4 + 2 3√ 4 + 3√ 42

= 4 + 2 3

√ 4 +

3√

42

8 − 4

= 4 + 2 3√ 4 +

3√ 424

= 1 +3√

4

2 +

3√

16

4

= 1 +3√

4

2 +

3√

2

2


Exercise:

Simplify the following. Rationalize the denominators.

24c−1/2d2/3

http://find/



1

24c / d /

18c−1/7d−3/5

2 (u1/3 + (uv)1/6 + v1/3)(u1/6 − v1/6)

3 3√ −84

4 4√

9x2

5

3√ 9a4b2

62√

5√ 8

+ 93√

16

7x2

−2x + 1

√ x + 1

81

3√

4 + 3√ −27


http://find/

Documents

A5 - Rational Exponents VJ