Upload
others
View
8
Download
0
Embed Size (px)
Citation preview
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
FractionsIntegral Exponents
Rational Exponents
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
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
dlsu-logo
FractionsIntegral Exponents
Rational Exponents
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