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dlsu-logo Fractions Integral Exponents Rational Exponents Fractions, Integral and Rational Exponents prepared by: Francis Joseph H. Campena Department of Mathematics De La Salle University - Manila Fractions prepared by: Francis Joseph H. Campena

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Page 1: Fractions, Integral and Rational Exponentsfrancisjosephcampena.weebly.com/Uploads/1/7/8/6/17869691/Fractions.pdf1. 4x 5y + 5y 4x 2. 2x 3 x + x +3 2x 6 3. x 1 x3 x 2 x2 x 3 x 4. y2

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FractionsIntegral Exponents

Rational Exponents

Fractions, Integral and RationalExponents

prepared by:

Francis Joseph H. Campena

Department of MathematicsDe La Salle University - Manila

Fractions prepared by: Francis Joseph H. Campena

Page 2: Fractions, Integral and Rational Exponentsfrancisjosephcampena.weebly.com/Uploads/1/7/8/6/17869691/Fractions.pdf1. 4x 5y + 5y 4x 2. 2x 3 x + x +3 2x 6 3. x 1 x3 x 2 x2 x 3 x 4. y2

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

Equivalent Fractions and Simplification ofFractions

A rational number of the formND

is defined as the ration oftwo integers, N,D where D 6= 0. In this form, N is calledthe numerator and D the denominator.If N,D are algebraic expressions, then

ND

is an algebraicfraction.However, we are more interested in a special type ofalgebraic fraction, where N,D are polynomials. These arecalled rational expressions.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

TheoremTwo fractions are said to be equivalent if and only if theyare numerals representing the same number, that is,

ab=

cd⇔ ad = bc; for b 6= 0,d 6= 0.

TheoremA fraction is said to be in lowest term if its numerator anddenominator do not have a common factor aside from 1.

acbc

=ab; for b 6= 0.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

Fractions follow these equivalances:

ab=−a−b

= −−ab

= − a−b

and

−ab=−ab

=a−b

= −−a−b

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

Reduce the following expressions in lowest terms.

1.32x5y6z92xy2z4

2.2x − 4x2 − 4

3.(x + 5)(x2 + x)

5x(5 + x)

4.x2 − 2x − 36 + x − x2

5.3 + x

3 + x(4 + x)

6.(2x + 1)(x2 − 4)(x + 2)(6x + 3)

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

Addition/Subtraction of Fractions

To get the sum or difference of two fractions with the samedenominator, we add or subtract the numerators, that is;

ab± c

b=

a± cb

.

To get the sum or difference of two fractions withdifferenct denominators, we first get the least commonmultiple of the denominators and add or subtract thecorresponding equivalent fractions, that is;

ab± c

bd=

ad ± cbbd

.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

Combine into a single fraction and simplify the result.

1.4x5y

+5y4x

2.2x

3− x+

x + 32x − 6

3.x − 1

x3 − x − 2x2 − x − 3

x

4.y2

y − 1− y2 + 1

y2 − 1

5.2

x − 3+

5x + 3

+6(x − 1)9− x2

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

Multiplication/Division of Fractions

To get the product of two fractions we just need to multiplytheir numerators and divide the result by the product ofthe denominators;

ab· c

d=

acbd

.

To get the quotient of two fractions, we need to multiplythe numerator by the reciprocal of the denominator;

ab÷ c

d=

ab· d

c=

adbc

.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

EXAMPLES

Perform the indicated operation and simplify the result.

1.3x5

8y3z· xz3

21y4z2 ·6y4

x3

2.5(x − 2y)

h3k2 ÷ x2 − 4y2

h2k5(x + 2y)

3.a2b3

z2 − 3z· z2 − 9

ab3 + ab2z÷ abz + 3ab

bz2 + z2

4.(

h − kh + k

+h + kh − k

)÷(

hh + k

+k

h − k

)

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

DefinitionA complex fraction is a fraction whose numeratorand/denominator contains another fraction.

ExampleThe following are examples of complex fractions:

2x − 1x−1

25x −

x−45x+1

and1

1− 1

1 +1

x − 1

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

Simplify the following complex fractions.

1.

2x− 3

y2x+

3y

2.x2 +

1x

x − 1 +1x

3.1 +

xy

1−(

xy

)2

4.x + 2 +

2x − 1

1 +2

x − 1

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations on FractionsComplex Fractions

Simplify the following complex fractionx − 1x + 2

+x − 3x + 1

x + 3x − 1

− x − 2x + 1

÷

x + 1x − 3

− x − 2x + 2

x − 1x + 3

− x + 1x − 2

.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

We recall the following laws of exponent and extend it toany integral exponents. For any integers n,m and any realnumber a,b we have

1) a0 = 1

2) a−n =1an

3) an · am = an+m

4) (an)m = anm

5)an

am = an−m,a 6= 0

6) (a · b)n = an · bn

7)(a

b

)n=

an

bn ;b 6= 0

Simplify the expression

(3x + 3)2(3x2 − 3)−1(x + 2)−1

3(x − 3)3(2x − 6)−1(x + 2)−2

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Simplify the following expressions.

1.−2x2y−4

5x−5y2

2.x−1 + y−1

(xy)−1

3.(

ab−4c2

a−3b5c−1

)−2

4.x−2y−1 + 5y−3

x−2 − 25x2y−4

5.(

2x+2y

2x

)x (2x−y

2−y

)y

6.p−3 + p−2z−1

p−2z−1 − z−3

7.(

am+1bm−1

ambm

)m

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Radicals

DefinitionIf a and b are real numbers and n is a positive integergreater than 1 such that bn = a, then b is called an nth

root of a.

Example

Since (3)2 = 9 and (−3)2 = 9, we say that 3,−3 aresquare roots of 9.Since (4)3 = 64, therefore we say that 4 is a cuberoot of 64.Since (−2)3 = −8, therefore we say that −2 is a cuberoot of −8.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

We note that the nth root of a number is not unique. Thus,we are lead to define an idea that produces a unique rootof a number, that is, the principal nth root.

DefinitionIf a and b are real numbers and n is a positive integergreater than 1, then the principal nth root of a denoted byn√

a is given by

n√

a =

{positiventh root of a if a > 0negativenth root of a if a < 0

.

Example√

9 = 3, 3√−64 = −4,

√16 = 4

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Considern√a

n : is called the order of the radical.√ : is called the radical sign.a : is called the radicand.

RemarkThe processes of converting a radical into an equivalentexpression in which no radical appears in a denominatoris called rationalizing the denominator.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Example

Example

1.√

57

2.3√

52 3√

4

3.4

3−√

2

4.x − 4√x − 2

5.x − 4√x − 2

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Radicals and Rational Exponents

Radicals can also be expressed using rational exponentsusing the law “(an)m = anm” and therefore we have a

1n is

an nth root of a. We consider the following definition.

DefinitionFor any positive integer n and any real number a, if n

√a is

a real number, then a1n = n√

a

ExampleConsider the following examples:

912 = 3;

(1

16

) 12

=14;64

13 = 3√

64 = 4

Fractions prepared by: Francis Joseph H. Campena

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Laws on Radicals

DefinitionFor any n,m ∈ Z+ which are relatively prime and anya,b ∈ R for which n

√a ∈ R, we have

1 amn =

(a

1n

)m=(

n√

a)m

= (am)1n =

(n√

am)

2(

n√

a)n

= a3 n√

ab = (ab)1n = (a)

1n (b)

1n = n√

a n√

b

4 n

√ab=

n√

an√

bwhere b 6= 0

5 m√

n√

a = nm√

a

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Examples

Express the following in radical form.

1. x− 43 x

56 x

23

2.8x− 3

5 y12

6x25 y

13

3.(

x−3ny2n

z−4n

)−2n

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Simplifying Radicals

A radical is in simplest form if it satisfies all of thefollowing conditions:(a) The radicand contains no factor that is a power having

an exponent greater than or equal to the index.(b) The radicand contains no fraction.(c) There is no radical sign in the denominator of a

fraction.(d) The index cannot be reduced any further.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Examples

Simplify the following radicals.

1. 4√

25x4

2. 3

√5x7y−27z2

3.4√

81x3y4√

4xy5

4.x

x −√

z

5.

√x − 4− 4

x

6.2

3−√

5

7.√

x +√

x + 1√

x −√

x + 1

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

Operations in Radicals

Radicals may be combined only if they are similar, that is,radials with the same index and radicand.

c n√

a± d n√

a = (c ± d) n√

a

To multiply radicals with the same index, we apply theformula (

c n√

a) (

d n√

b)= cd n

√ab

To multiply radicals with different indices, convert eachradical into an equivalent radical form having the sameindex and apply the above formula.

Fractions prepared by: Francis Joseph H. Campena

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FractionsIntegral Exponents

Rational Exponents

ExamplesPerform the indicated operations.1. 2 5√

7− 6x 5√

72. 4√

3x − 4√

48x3. 2√

27− 2√

18− 3√

48− 3√

504.√

16x − 16−√

9x3 − 9x2

5.√

3(√

6−√

10)6.√

a− b√

a + b7.√

x√

2x + 5y8. 3√

4 4√

39. (3−

√5)(√

5− 3)

10.

(√5−√

23

)2

Fractions prepared by: Francis Joseph H. Campena

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EXERCISES

Simplify the following:

1.√

x + 1x − 1

+ 2

√1− 1

x2 −

√(x − 1

x

)1x

2.

√14+

19√

14+

√19

3.

√a2 − b2 +

2b2√

a2 − b2

a2 + b2

Fractions prepared by: Francis Joseph H. Campena