L5 Radical Expressions

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

Outline

1 Radical Expressionsnth root of aSimplification of Radical ExpressionsOperations Involving Radical Expressions

Addition and Subtraction of Radical ExpressionsMultiplication and Division of Radical Expressions

Rationalizing Radical Expressions


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.


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.


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.

Note:

1. The principal nth root is unique.

2. If n is even, the nth roots of negative real numbers are not realnumbers.



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 realnumbers.


Radical Expression

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


Radical Expression



Radical Expression



Radical Expression



Radical Expression




Examples:

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

That is,√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

2011√0 = 0

√−9 is not a real number because there is no real number b such

that b2 = −9.



Examples:

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

√16 = 4.


√27 = 3.

3√−27 = −3

2011√0 = 0


that b2 = −9.



Examples:


√16 = 4.


√27 = 3.

3√−27 = −3

2011√0 = 0


that b2 = −9.



Examples:


√16 = 4.


√27 = 3.

3√−27

= −32011√0 = 0


that b2 = −9.



Examples:


√16 = 4.


√27 = 3.

3√−27 = −3

2011√0 = 0


that b2 = −9.



Examples:


√16 = 4.


√27 = 3.

3√−27 = −3

2011√0

= 0√−9 is not a real number because there is no real number b such

that b2 = −9.



Examples:


√16 = 4.


√27 = 3.

3√−27 = −3

2011√0 = 0


that b2 = −9.



Examples:


√16 = 4.


√27 = 3.

3√−27 = −3

2011√0 = 0

√−9

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



Examples:


√16 = 4.


√27 = 3.

3√−27 = −3

2011√0 = 0


that b2 = −9.


Rational Exponents for Radical Expressions

Definition

If n is a positive integer and m is an integer such that mn is in lowest

terms, thena

1n = n

√a, and

amn =

(a

1n

)m= ( 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


Rational Exponents for Radical Expressions

Definition

If n is a positive integer and m is an integer such that mn is in lowest

terms, thena

1n = n

√a, and

amn =

(a

1n

)m= ( 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


Radical Expressions and Rational Exponents

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

Laws of Rational Exponents

Let a, b ∈ R; m,n, p, q ∈ Z; n, q 6= 0:

amn · a

pq = a

mn+ p

q

amn

apq

= amn− p

q , a 6= 0

a−mn =

1

amn

, a 6= 0

(a

mn

) pq = a

mpnq

amn · b

mn = (ab)

mn

amn

bmn

=(ab

)mn, b 6= 0


Radical Expression and the Rational Exponents

Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3 + a−1/6 − a−1/3

= a1/3 +1

a1/6− 1

a1/3



Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3 + a−1/6 − a−1/3

= a1/3 +1

a1/6− 1

a1/3



Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3 + a−1/6 − a−1/3

= a1/3 +1

a1/6− 1

a1/3



Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3 + a−1/6 − a−1/3

= a1/3 +1

a1/6− 1

a1/3



Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3 + a−1/6 − a−1/3

= a1/3 +1

a1/6− 1

a1/3



Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3

+ a−1/6 − a−1/3

= a1/3 +1

a1/6− 1

a1/3



Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3 + a−1/6

− a−1/3

= a1/3 +1

a1/6− 1

a1/3



Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3 + a−1/6 − a−1/3

= a1/3 +1

a1/6− 1

a1/3



Examples:

1. (x1/2 + y1/2)(x1/2 − y1/2)

= x1/2x1/2 + x1/2y1/2 − x1/2y1/2 − y1/2y1/2

= x− y

2.a+ a1/2 − a1/3

a2/3

=a

a2/3+

a1/2

a2/3− a1/3

a2/3

= a1/3 + a−1/6 − a−1/3

= a1/3 +1

a1/6− 1

a1/3


Simplification of Radical Expressions

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√b

n

√a

b=

n√a

n√b, b 6= 0

n√am = am/n

n√

m√a = mn

√a



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√b

n

√a

b=

n√a

n√b, b 6= 0

n√am = am/n

n√

m√a = mn

√a



Examples:

1.√2 ·√18 =

√2 · 18 =

√36 = 6

2.3√54

3√2

= 3

√54

2= 3√27 = 3

3.4√x3 = x3/4 (x ≥ 0 if the index is even.)

Note: if n is even, am/n = n√am is true only if a ≥ 0.

Example: 43/2 = (√4)3 = 23 = 8 and

√43 =

√64 = 8

(−1)6/2 = (−1)3 = −1 but√(−1)6 =

√1 = 1



Examples:

1.√2 ·√18 =

√2 · 18 =

√36 = 6

2.3√54

3√2

= 3

√54

2= 3√27 = 3



Example: 43/2 = (√4)3 = 23 = 8 and

√43 =

√64 = 8

(−1)6/2 = (−1)3 = −1 but√(−1)6 =

√1 = 1



Examples:

1.√2 ·√18 =

√2 · 18 =

√36 = 6

2.3√54

3√2

= 3

√54

2= 3√27 = 3



Example: 43/2 = (√4)3 = 23 = 8 and

√43 =

√64 = 8

(−1)6/2 = (−1)3 = −1 but√

(−1)6 =√1 = 1



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


Example:√

18x3y2

=√32 · 2 · x2 · x · y2

=√32 ·√2 ·√x2 ·√x ·√y2

= 3xy√2x




Example:√

18x3y2

=√32 · 2 · x2 · x · y2

=√32 ·√2 ·√x2 ·√x ·√y2

= 3xy√2x




Example:√

18x3y2

=√32 · 2 · x2 · x · y2

=√32 ·√2 ·√x2 ·√x ·√

y2

= 3xy√2x




Example:√

18x3y2

=√32 · 2 · x2 · x · y2

=√32 ·√2 ·√x2 ·√x ·√

y2

= 3xy√2x




Example: 3√−8x7y2

= 3√

(−2)3 · x6 · x · y2= 3

√(−2)3 · 3

√x6 · 3√x · 3√y2

= −2x2 3√xy2





= 3√(−2)3 · x6 · x · y2

= 3√

(−2)3 · 3√x6 · 3√x · 3√y2

= −2x2 3√xy2





= 3√(−2)3 · x6 · x · y2

= 3√(−2)3 · 3

√x6 · 3√x · 3√y2

= −2x2 3√xy2





= 3√(−2)3 · x6 · x · y2

= 3√(−2)3 · 3

√x6 · 3√x · 3√y2

= −2x2 3√xy2




Example:3

√10y4

3x2

=3

√10y4

3x2· 3

2x

32x=

3

√90xy4

33 · x3=

3√90xy · y33√33 · x3

=y 3√90xy

3x




Example:3

√10y4

3x2

=3

√10y4

3x2· 3

2x

32x

=3

√90xy4

33 · x3=

3√90xy · y33√33 · x3

=y 3√90xy

3x




Example:3

√10y4

3x2

=3

√10y4

3x2· 3

2x

32x=

3

√90xy4

33 · x3

=3√90xy · y33√33 · x3

=y 3√90xy

3x




Example:3

√10y4

3x2

=3

√10y4

3x2· 3

2x

32x=

3

√90xy4

33 · x3=

3√

90xy · y33√33 · x3

=y 3√90xy

3x




Example:3

√10y4

3x2

=3

√10y4

3x2· 3

2x

32x=

3

√90xy4

33 · x3=

3√

90xy · y33√33 · x3

=y 3√90xy

3x



Example: 4√9

=4√32 = 32/4 = 31/2 =

√3

Example:6√16x4 = 6

√(2x)4 = 3

√(2x)2 =

3√4x2



Example: 4√9 =

4√32

= 32/4 = 31/2 =√3


√(2x)4 = 3

√(2x)2 =

3√4x2



Example: 4√9 =

4√32 = 32/4

= 31/2 =√3


√(2x)4 = 3

√(2x)2 =

3√4x2



Example: 4√9 =

4√32 = 32/4 = 31/2

=√3


√(2x)4 = 3

√(2x)2 =

3√4x2



Example: 4√9 =

4√32 = 32/4 = 31/2 =

√3


√(2x)4 = 3

√(2x)2 =

3√4x2



Example: 4√9 =

4√32 = 32/4 = 31/2 =

√3

Example:6√16x4

= 6√(2x)4 = 3

√(2x)2 =

3√4x2



Example: 4√9 =

4√32 = 32/4 = 31/2 =

√3


√(2x)4

= 3√(2x)2 =

3√4x2



Example: 4√9 =

4√32 = 32/4 = 31/2 =

√3


√(2x)4 = 3

√(2x)2

=3√4x2



Example: 4√9 =

4√32 = 32/4 = 31/2 =

√3


√(2x)4 = 3

√(2x)2 =

3√4x2


Addition and Subtraction of Radical Expressions

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

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




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




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




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




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




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




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




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


Multiplication and Division of Radical Expressions

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

=

√15x · 2x√6x3

=

√30x2√6x3

=

√30x2

6x3=

√5

x· xx

=

√5x

x




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· xx

=

√5x

x




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· xx

=

√5x

x




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· xx

=

√5x

x




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· xx

=

√5x

x




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· xx

=

√5x

x




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· xx

=

√5x

x




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· xx

=

√5x

x




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· xx

=

√5x

x




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· xx

=

√5x

x



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

LCM of indices: 4 √2 · 4√8 =

4√22 · 4√8

= 4√4(8)

= 4√32

=4√25

= 2 4√2




√a = nm

√am.


LCM of indices: 4

√2 · 4√8 =

4√22 · 4√8

= 4√4(8)

= 4√32

=4√25

= 2 4√2




√a = nm

√am.



4√22 · 4√8

= 4√4(8)

= 4√32

=4√25

= 2 4√2




√a = nm

√am.



4√22 · 4√8

= 4√

4(8)

= 4√32

=4√25

= 2 4√2




√a = nm

√am.



4√22 · 4√8

= 4√

4(8)

= 4√32

=4√25

= 2 4√2




√a = nm

√am.



4√22 · 4√8

= 4√

4(8)

= 4√32

=4√25

= 2 4√2




√a = nm

√am.



4√22 · 4√8

= 4√

4(8)

= 4√32

=4√25

= 2 4√2


Rationalizing a Denominator

If denominator consists of one radical

Multiply both the numerator and denominator by an expression that willmake the radicand of the denominator a perfect power of the index.

Example:

√16x4

3z

=4x2√3z·√3z√3z

=4x2√3z√

32z2

=4x2√3z

3z





Example:

√16x4

3z

=4x2√3z

·√3z√3z

=4x2√3z√

32z2

=4x2√3z

3z





Example:

√16x4

3z

=4x2√3z·√3z√3z

=4x2√3z√

32z2

=4x2√3z

3z





Example:

√16x4

3z

=4x2√3z·√3z√3z

=4x2√3z√

32z2

=4x2√3z

3z





Example:

√16x4

3z

=4x2√3z·√3z√3z

=4x2√3z√

32z2

=4x2√3z

3z



If denominator consists of two or more radicals

To rationalize the denominator, use the special products:

(x− y)(x+ y) = x2 − y2

(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





(x− y)(x+ y) = x2 − y2

(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





(x− y)(x+ y) = x2 − y2

(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





(x− y)(x+ y) = x2 − y2

(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





(x− y)(x+ y) = x2 − y2

(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





(x− y)(x+ y) = x2 − y2

(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





(x− y)(x+ y) = x2 − y2

(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





(x− y)(x+ y) = x2 − y2

(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



Example:1

2− 3√4

=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

= 1 +3√4

2+

3√16

4

= 1 +3√4

2+

3√2

2



Example:1

2− 3√4

=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

= 1 +3√4

2+

3√16

4

= 1 +3√4

2+

3√2

2



Example:1

2− 3√4

=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

= 1 +3√4

2+

3√16

4

= 1 +3√4

2+

3√2

2



Example:1

2− 3√4

=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

= 1 +3√4

2+

3√16

4

= 1 +3√4

2+

3√2

2



Example:1

2− 3√4

=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

= 1 +3√4

2+

3√16

4

= 1 +3√4

2+

3√2

2



Example:1

2− 3√4

=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

= 1 +3√4

2+

3√16

4

= 1 +3√4

2+

3√2

2


Exercise:

Simplify the following. Rationalize the denominators.

124c−1/2d2/3

18c−1/7d−3/5

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

33√−84

44√9x2

5

√3√9a4b2

62√5√8

+9

3√16

7x2 − 2x+ 1√

x+ 1

81

3√4 + 3√−27


Documents

L5 Radical Expressions