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The Real Number System and Integer Exponents
Mathematics 17
Institute of Mathematics, University of the Philippines-Diliman
Lecture 2
Math 17 (UP-IMath) R and Integer Exponents Lec 2 1 / 33
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Outline
1 The Real Number SystemAxioms on ROrder in RThe Real Number LineInterval NotationThe Absolute Value
2 Integer ExponentsLaws of Exponents
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The Real Number System
Recall: R, the set of real numbers
The real number system consists of R and two operations on its elements:
addition and multiplication
Operation Symbol ResultAddition + Sum a + b
Multiplication · Product a · b
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The Real Number System
Recall: R, the set of real numbers
The real number system consists of R and two operations on its elements:
addition and multiplication
Operation Symbol ResultAddition + Sum a + b
Multiplication · Product a · b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 3 / 33
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Axioms on Equality
Axioms: logical statements that are assumed to be true
For any real numbers a, b, c,
Reflexive : a = a
Symmetric : If a = b, then b = a.
Transitive : If a = b and b = c, then a = c.
Additive : If a = b, then a + c = b + c.
Multiplicative : If a = b, then ac = bc.
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Axioms on Equality
Axioms: logical statements that are assumed to be true
For any real numbers a, b, c,
Reflexive : a = a
Symmetric : If a = b, then b = a.
Transitive : If a = b and b = c, then a = c.
Additive : If a = b, then a + c = b + c.
Multiplicative : If a = b, then ac = bc.
Math 17 (UP-IMath) R and Integer Exponents Lec 2 4 / 33
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Axioms for Addition and Multiplication
For any real numbers a, b and c
Axiom Addition Multiplication
Closure a + b ∈ R a · b ∈ RAssociativity (a + b) + c = a + (b + c) (a · b) · c = a · (b · c)
Commutativity a + b = b + a a · b = b · a
Distributivity: c · (a + b) = c · a + c · b
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Let a ∈ R.
Identity Axiom for Addition :There exists a real number, zero (0), such thata + 0 = 0 + a = a
0 is the identity element for addition.
Identity Axiom for Multiplication :There exists a real number, one (1), such thata · 1 = 1 · a = a
1 is the identity element for multiplication.
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Let a ∈ R.
Inverse Axiom for Addition :There exists a real number, −a, such thata + (−a) = (−a) + a = 0−a is the additive inverse of a.
Inverse Axiom for Multiplication :
If a 6= 0, there exists a real number,1a
, such that
a · 1a
=1a· a = 1
1a
is the multiplicative inverse of a.
Math 17 (UP-IMath) R and Integer Exponents Lec 2 7 / 33
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Cancellation Laws
For any a, b, c ∈ R,
Addition :If a + c = b + c, then a = b.If c + a = c + b, then a = b.
Multiplication :If ac = bc, c 6= 0, then a = b.If ca = cb, c 6= 0, then a = b.
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For any real numbers a and b,
a · 0 = 0
If ab = 0, then a = 0 or b = 0
−(−a) = a
(−a)b = −ab
(−a)(−b) = ab
−(a + b) = (−a) + (−b)
(−1)a = −a
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Subtraction
Definition
If a, b ∈ R, then subtraction is the operation that assigns to a and b a realnumber, a− b, the difference of a and b, where
a− b = a + (−b)
.
For any a, b, c ∈ R,
a− a = 0a− (−b) = a + b
a(b− c) = ab− ac
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Subtraction
Definition
If a, b ∈ R, then subtraction is the operation that assigns to a and b a realnumber, a− b, the difference of a and b, where
a− b = a + (−b)
.
For any a, b, c ∈ R,
a− a = 0a− (−b) = a + b
a(b− c) = ab− ac
Math 17 (UP-IMath) R and Integer Exponents Lec 2 10 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0
a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0
a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Division
Definition
If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b
a real number, a÷ b =a
b, the quotient of a and b, where
a
b= a · 1
b.
For any a, b, c, d ∈ R, with b, d 6= 0a
a= 1 if a 6= 0
a
1= a
1(1a
) = a for a 6= 0
1(ab
) =b
afor a 6= 0
a
b=
c
d⇔ ad = bc
a
b=
ac
bcif c 6= 0
−a
b=
a
−b= −a
b
−a
−b=
a
b
Math 17 (UP-IMath) R and Integer Exponents Lec 2 11 / 33
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Other properties of division: For a, b, c, d ∈ R, b, d 6= 0
Sum :a
b+
c
d=
ad + bc
bd
Difference :a
b− c
d=
ad− bc
bd
Product :a
b· cd
=ac
bd
Quotient :a
b÷ c
d=
a
b· dc
=ad
bcprovided c 6= 0
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Order Axioms of R
For any real number a,
a is positive if and only if a > 0.
a is negative if and only if a < 0.
For any real numbers a, b, c
Trichotomy One and only one of the following relations holds:
a = b, a > b or a < b
Transitive If a > b and b > c then a > c.
Addition If a > b, then a + c > b + c.
Multiplication If a > b and c > 0, then ac > bc.
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Order Axioms of R
For any real number a,
a is positive if and only if a > 0.
a is negative if and only if a < 0.
For any real numbers a, b, c
Trichotomy One and only one of the following relations holds:
a = b, a > b or a < b
Transitive If a > b and b > c then a > c.
Addition If a > b, then a + c > b + c.
Multiplication If a > b and c > 0, then ac > bc.
Math 17 (UP-IMath) R and Integer Exponents Lec 2 13 / 33
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For any a, b, c ∈ RThe set of positive real numbers is closed under addition andmultiplication.
If a > b, then −a < −b.
a2 ≥ 0If a > b and c < 0, then ac < bc.
If a > 0, then1a
> 0.
If a > b > 0, then1a
<1b
.
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For any a, b, c ∈ RThe set of positive real numbers is closed under addition andmultiplication.
If a > b, then −a < −b.
a2 ≥ 0If a > b and c < 0, then ac < bc.
If a > 0, then1a
> 0.
If a > b > 0, then1a
<1b
.
Math 17 (UP-IMath) R and Integer Exponents Lec 2 14 / 33
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For any a, b, c ∈ RThe set of positive real numbers is closed under addition andmultiplication.
If a > b, then −a < −b.
a2 ≥ 0
If a > b and c < 0, then ac < bc.
If a > 0, then1a
> 0.
If a > b > 0, then1a
<1b
.
Math 17 (UP-IMath) R and Integer Exponents Lec 2 14 / 33
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For any a, b, c ∈ RThe set of positive real numbers is closed under addition andmultiplication.
If a > b, then −a < −b.
a2 ≥ 0If a > b and c < 0, then ac < bc.
If a > 0, then1a
> 0.
If a > b > 0, then1a
<1b
.
Math 17 (UP-IMath) R and Integer Exponents Lec 2 14 / 33
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For any a, b, c ∈ RThe set of positive real numbers is closed under addition andmultiplication.
If a > b, then −a < −b.
a2 ≥ 0If a > b and c < 0, then ac < bc.
If a > 0, then1a
> 0.
If a > b > 0, then1a
<1b
.
Math 17 (UP-IMath) R and Integer Exponents Lec 2 14 / 33
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For any a, b, c ∈ RThe set of positive real numbers is closed under addition andmultiplication.
If a > b, then −a < −b.
a2 ≥ 0If a > b and c < 0, then ac < bc.
If a > 0, then1a
> 0.
If a > b > 0, then1a
<1b
.
Math 17 (UP-IMath) R and Integer Exponents Lec 2 14 / 33
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The Real Number Line
Math 17 (UP-IMath) R and Integer Exponents Lec 2 15 / 33
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The Real Number Line
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The Real Number Line
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The Real Number Line
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The Real Number Line
There is a one-to-one correspondence between
the points on the line l and R.
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The Real Number Line
All real numbers can be put in sequence
on a line.
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Interval Notation
Let a, b ∈ R such that a < b.
The open interval (a, b) is the set {x ∈ R | a < x < b}.
The closed interval [a, b], is the open interval (a, b) together with itstwo endpoints.
a is called the left endpoint
b is called the right endpoint
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Interval Notation
Let a, b ∈ R such that a < b.
The open interval (a, b) is the set {x ∈ R | a < x < b}.
The closed interval [a, b], is the open interval (a, b) together with itstwo endpoints.
a is called the left endpoint
b is called the right endpoint
Math 17 (UP-IMath) R and Integer Exponents Lec 2 16 / 33
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Interval Notation
Let a, b ∈ R such that a < b.
The open interval (a, b) is the set {x ∈ R | a < x < b}.
The closed interval [a, b], is the open interval (a, b) together with itstwo endpoints.
a is called the left endpoint
b is called the right endpoint
Math 17 (UP-IMath) R and Integer Exponents Lec 2 16 / 33
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Interval Notation
Let a, b ∈ R such that a < b.
The open interval (a, b) is the set {x ∈ R | a < x < b}.
The closed interval [a, b], is the open interval (a, b) together with itstwo endpoints.
a is called the left endpoint
b is called the right endpoint
Math 17 (UP-IMath) R and Integer Exponents Lec 2 16 / 33
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Interval Notation
Let a, b ∈ R such that a < b.
The open interval (a, b) is the set {x ∈ R | a < x < b}.
The closed interval [a, b], is the open interval (a, b) together with itstwo endpoints.
a is called the left endpoint
b is called the right endpoint
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Interval Notation Set Notation Number Line Graph
[a, b] {x | a ≤ x ≤ b}
(a, b) {x | a < x < b}
[a, b) {x | a ≤ x < b}
(a, b] {x | a < x ≤ b}
[a, +∞) {x | x ≥ a}
(a, +∞) {x | x > a}
(−∞, b] {x | x ≤ b}
(−∞, b) {x | x < b}
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Example.
1 [−4, 1] ∪ (−2, 3]
= [−4, 3]2 [−4, 1] ∩ (−2, 3]
= (−2, 1]
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Example.
1 [−4, 1] ∪ (−2, 3]
= [−4, 3]2 [−4, 1] ∩ (−2, 3]
= (−2, 1]
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Example.
1 [−4, 1] ∪ (−2, 3]
= [−4, 3]2 [−4, 1] ∩ (−2, 3]
= (−2, 1]
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Example.
1 [−4, 1] ∪ (−2, 3]
= [−4, 3]2 [−4, 1] ∩ (−2, 3]
= (−2, 1]
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Example.
1 [−4, 1] ∪ (−2, 3]
= [−4, 3]2 [−4, 1] ∩ (−2, 3]
= (−2, 1]
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Example.
1 [−4, 1] ∪ (−2, 3] = [−4, 3]
2 [−4, 1] ∩ (−2, 3]
= (−2, 1]
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Example.
1 [−4, 1] ∪ (−2, 3] = [−4, 3]2 [−4, 1] ∩ (−2, 3]
= (−2, 1]
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Example.
1 [−4, 1] ∪ (−2, 3] = [−4, 3]2 [−4, 1] ∩ (−2, 3]
= (−2, 1]
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Example.
1 [−4, 1] ∪ (−2, 3] = [−4, 3]2 [−4, 1] ∩ (−2, 3] = (−2, 1]
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The Absolute Value
If x is any real number, the absolute value of x, written as |x|, is definedas:
|x| =
−x if x < 00 if x = 0x if x > 0
Example. | − 7| = |7| = 7
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The Absolute Value
If x is any real number, the absolute value of x, written as |x|, is definedas:
|x| =
−x if x < 00 if x = 0x if x > 0
Example. | − 7| = |7| = 7
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Let x, y ∈ R. Then
|x| ≥ 0| − x| = |x||xy| = |x| · |y|∣∣∣∣xy∣∣∣∣ = |x||y| , y 6= 0
Example:
∣∣∣∣3− 95
∣∣∣∣ = ∣∣∣∣−65
∣∣∣∣ = | − 6||5|
=65
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Let x, y ∈ R. Then
|x| ≥ 0| − x| = |x||xy| = |x| · |y|∣∣∣∣xy∣∣∣∣ = |x||y| , y 6= 0
Example:
∣∣∣∣3− 95
∣∣∣∣ = ∣∣∣∣−65
∣∣∣∣ = | − 6||5|
=65
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Geometric Interpretation of Absolute Value
|x| represents the distance of the point corresponding to x from the pointcorresponding to 0.
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Geometric Interpretation of Absolute Value
|x| represents the distance of the point corresponding to x from the pointcorresponding to 0.
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Geometric Interpretation of Absolute Value
|x| represents the distance of the point corresponding to x from the pointcorresponding to 0.
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Let a > 0|x| = a if and only if x = ±a
|x| < a if and only if − a < x < a
|x| > a if and only if x < −a or x > a
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Let a > 0|x| = a if and only if x = ±a
|x| < a if and only if − a < x < a
|x| > a if and only if x < −a or x > a
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Let a > 0|x| = a if and only if x = ±a
|x| < a if and only if − a < x < a
|x| > a if and only if x < −a or x > a
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Let a > 0|x| = a if and only if x = ±a
|x| < a if and only if − a < x < a
|x| > a if and only if x < −a or x > a
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Let a > 0|x| = a if and only if x = ±a
|x| < a if and only if − a < x < a
|x| > a if and only if x < −a or x > a
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Distance between Real Numbers
The distance between two real numbers x and y is
d = |x− y| = |y − x|
Example: The distance between −11 and 5 is
| − 11− 5| = | − 16| = 16
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Distance between Real Numbers
The distance between two real numbers x and y is
d = |x− y| = |y − x|
Example: The distance between −11 and 5 is
| − 11− 5| = | − 16| = 16
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Distance between Real Numbers
The distance between two real numbers x and y is
d = |x− y| = |y − x|
Example: The distance between −11 and 5 is
| − 11− 5| = | − 16| = 16
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For x, y ∈ R,
|x− y| ≥ 0|x− y| = 0 if and only if x = y
Triangle Inequality: |x + y| ≤ |x|+ |y|
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Integer Exponents
Let a ∈ R, n ∈ Z. The nth power of a is denoted by an.
If n > 0, then an = a · a · a · · · · · a︸ ︷︷ ︸n times
.
If a 6= 0, then a0 = 1 and a−1 =1a
.
If n > 0 and a 6= 0, then a−n =1an
.
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Integer Exponents
Let a ∈ R, n ∈ Z. The nth power of a is denoted by an.
If n > 0, then an = a · a · a · · · · · a︸ ︷︷ ︸n times
.
If a 6= 0, then a0 = 1 and a−1 =1a
.
If n > 0 and a 6= 0, then a−n =1an
.
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Integer Exponents
Let a ∈ R, n ∈ Z. The nth power of a is denoted by an.
If n > 0, then an = a · a · a · · · · · a︸ ︷︷ ︸n times
.
If a 6= 0, then a0 = 1
and a−1 =1a
.
If n > 0 and a 6= 0, then a−n =1an
.
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Integer Exponents
Let a ∈ R, n ∈ Z. The nth power of a is denoted by an.
If n > 0, then an = a · a · a · · · · · a︸ ︷︷ ︸n times
.
If a 6= 0, then a0 = 1 and a−1 =1a
.
If n > 0 and a 6= 0, then a−n =1an
.
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Integer Exponents
Let a ∈ R, n ∈ Z. The nth power of a is denoted by an.
If n > 0, then an = a · a · a · · · · · a︸ ︷︷ ︸n times
.
If a 6= 0, then a0 = 1 and a−1 =1a
.
If n > 0 and a 6= 0, then a−n =1an
.
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Laws of Exponents: Product Law
Let n, m ∈ Z, a ∈ R, then an · am = an+m.
Examples:
42 · 43 = 42+3
= 45
= 1024
x2 · x−5 = x−3
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Laws of Exponents: Product Law
Let n, m ∈ Z, a ∈ R, then an · am = an+m.
Examples:
42 · 43 = 42+3
= 45
= 1024
x2 · x−5 = x−3
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Laws of Exponents: Product Law
Let n, m ∈ Z, a ∈ R, then an · am = an+m.
Examples:
42 · 43 =
42+3
= 45
= 1024
x2 · x−5 = x−3
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Laws of Exponents: Product Law
Let n, m ∈ Z, a ∈ R, then an · am = an+m.
Examples:
42 · 43 = 42+3
= 45
= 1024
x2 · x−5 = x−3
Math 17 (UP-IMath) R and Integer Exponents Lec 2 26 / 33
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Laws of Exponents: Product Law
Let n, m ∈ Z, a ∈ R, then an · am = an+m.
Examples:
42 · 43 = 42+3
= 45
= 1024
x2 · x−5 = x−3
Math 17 (UP-IMath) R and Integer Exponents Lec 2 26 / 33
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Laws of Exponents: Product Law
Let n, m ∈ Z, a ∈ R, then an · am = an+m.
Examples:
42 · 43 = 42+3
= 45
= 1024
x2 · x−5 = x−3
Math 17 (UP-IMath) R and Integer Exponents Lec 2 26 / 33
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Laws of Exponents: Product Law
Let n, m ∈ Z, a ∈ R, then an · am = an+m.
Examples:
42 · 43 = 42+3
= 45
= 1024
x2 · x−5 =
x−3
Math 17 (UP-IMath) R and Integer Exponents Lec 2 26 / 33
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Laws of Exponents: Product Law
Let n, m ∈ Z, a ∈ R, then an · am = an+m.
Examples:
42 · 43 = 42+3
= 45
= 1024
x2 · x−5 = x−3
Math 17 (UP-IMath) R and Integer Exponents Lec 2 26 / 33
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 = y−6
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 = y−6
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 =
23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 = y−6
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 = y−6
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 = y−6
Math 17 (UP-IMath) R and Integer Exponents Lec 2 27 / 33
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 = y−6
Math 17 (UP-IMath) R and Integer Exponents Lec 2 27 / 33
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 =
x−3
(y2)−3 = y−6
Math 17 (UP-IMath) R and Integer Exponents Lec 2 27 / 33
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 = y−6
Math 17 (UP-IMath) R and Integer Exponents Lec 2 27 / 33
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 =
y−6
Math 17 (UP-IMath) R and Integer Exponents Lec 2 27 / 33
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Laws of Exponents: Power raised to a Power Law
Let n, m ∈ Z, a ∈ R, then (an)m = anm.
Examples:
(23)4 = 23·4
= 212
= 4096
(x−1)3 = x−3
(y2)−3 = y−6
Math 17 (UP-IMath) R and Integer Exponents Lec 2 27 / 33
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Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9= 36
25 · 55 = (2 · 5)5
= 105
= 100000
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Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9= 36
25 · 55 = (2 · 5)5
= 105
= 100000
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Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 =
(−2)2 · (3)2
= 4 · 9= 36
25 · 55 = (2 · 5)5
= 105
= 100000
Math 17 (UP-IMath) R and Integer Exponents Lec 2 28 / 33
![Page 96: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/96.jpg)
Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9= 36
25 · 55 = (2 · 5)5
= 105
= 100000
Math 17 (UP-IMath) R and Integer Exponents Lec 2 28 / 33
![Page 97: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/97.jpg)
Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9
= 36
25 · 55 = (2 · 5)5
= 105
= 100000
Math 17 (UP-IMath) R and Integer Exponents Lec 2 28 / 33
![Page 98: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/98.jpg)
Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9= 36
25 · 55 = (2 · 5)5
= 105
= 100000
Math 17 (UP-IMath) R and Integer Exponents Lec 2 28 / 33
![Page 99: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/99.jpg)
Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9= 36
25 · 55 =
(2 · 5)5
= 105
= 100000
Math 17 (UP-IMath) R and Integer Exponents Lec 2 28 / 33
![Page 100: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/100.jpg)
Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9= 36
25 · 55 = (2 · 5)5
= 105
= 100000
Math 17 (UP-IMath) R and Integer Exponents Lec 2 28 / 33
![Page 101: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/101.jpg)
Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9= 36
25 · 55 = (2 · 5)5
= 105
= 100000
Math 17 (UP-IMath) R and Integer Exponents Lec 2 28 / 33
![Page 102: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/102.jpg)
Laws of Exponents: Power of a Product Law
Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.
Examples:
(−2 · 3)2 = (−2)2 · (3)2
= 4 · 9= 36
25 · 55 = (2 · 5)5
= 105
= 100000
Math 17 (UP-IMath) R and Integer Exponents Lec 2 28 / 33
![Page 103: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/103.jpg)
Laws of Exponents: Quotient Law
Let n, m ∈ Z, a ∈ R. If a 6= 0, thenan
am= an−m.
Examples:
x10
x6= x10−6 = x4
y2
y−7= y2−(−7) = y9
1110
1110= 110 = 1
Math 17 (UP-IMath) R and Integer Exponents Lec 2 29 / 33
![Page 104: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/104.jpg)
Laws of Exponents: Quotient Law
Let n, m ∈ Z, a ∈ R. If a 6= 0, thenan
am= an−m.
Examples:
x10
x6= x10−6 = x4
y2
y−7= y2−(−7) = y9
1110
1110= 110 = 1
Math 17 (UP-IMath) R and Integer Exponents Lec 2 29 / 33
![Page 105: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/105.jpg)
Laws of Exponents: Quotient Law
Let n, m ∈ Z, a ∈ R. If a 6= 0, thenan
am= an−m.
Examples:
x10
x6= x10−6 = x4
y2
y−7= y2−(−7) = y9
1110
1110= 110 = 1
Math 17 (UP-IMath) R and Integer Exponents Lec 2 29 / 33
![Page 106: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/106.jpg)
Laws of Exponents: Quotient Law
Let n, m ∈ Z, a ∈ R. If a 6= 0, thenan
am= an−m.
Examples:
x10
x6= x10−6 = x4
y2
y−7= y2−(−7) = y9
1110
1110= 110 = 1
Math 17 (UP-IMath) R and Integer Exponents Lec 2 29 / 33
![Page 107: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/107.jpg)
Laws of Exponents: Quotient Law
Let n, m ∈ Z, a ∈ R. If a 6= 0, thenan
am= an−m.
Examples:
x10
x6= x10−6 = x4
y2
y−7= y2−(−7) = y9
1110
1110= 110 = 1
Math 17 (UP-IMath) R and Integer Exponents Lec 2 29 / 33
![Page 108: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/108.jpg)
Laws of Exponents: Power of a Quotient Law
Let n ∈ Z, a, b ∈ R. If b 6= 0, then(a
b
)n=
an
bn.
Examples.
(x
y
)4
=x4
y4
63
183=
(618
)3
=(
13
)3
=127
Math 17 (UP-IMath) R and Integer Exponents Lec 2 30 / 33
![Page 109: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/109.jpg)
Laws of Exponents: Power of a Quotient Law
Let n ∈ Z, a, b ∈ R. If b 6= 0, then(a
b
)n=
an
bn.
Examples.
(x
y
)4
=x4
y4
63
183=
(618
)3
=(
13
)3
=127
Math 17 (UP-IMath) R and Integer Exponents Lec 2 30 / 33
![Page 110: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/110.jpg)
Laws of Exponents: Power of a Quotient Law
Let n ∈ Z, a, b ∈ R. If b 6= 0, then(a
b
)n=
an
bn.
Examples.
(x
y
)4
=x4
y4
63
183=
(618
)3
=(
13
)3
=127
Math 17 (UP-IMath) R and Integer Exponents Lec 2 30 / 33
![Page 111: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/111.jpg)
Laws of Exponents: Power of a Quotient Law
Let n ∈ Z, a, b ∈ R. If b 6= 0, then(a
b
)n=
an
bn.
Examples.
(x
y
)4
=x4
y4
63
183=
(618
)3
=(
13
)3
=127
Math 17 (UP-IMath) R and Integer Exponents Lec 2 30 / 33
![Page 112: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/112.jpg)
Laws of Exponents: Power of a Quotient Law
Let n ∈ Z, a, b ∈ R. If b 6= 0, then(a
b
)n=
an
bn.
Examples.
(x
y
)4
=x4
y4
63
183=
(618
)3
=(
13
)3
=127
Math 17 (UP-IMath) R and Integer Exponents Lec 2 30 / 33
![Page 113: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/113.jpg)
Laws of Exponents: Power of a Quotient Law
Let n ∈ Z, a, b ∈ R. If b 6= 0, then(a
b
)n=
an
bn.
Examples.
(x
y
)4
=x4
y4
63
183=
(618
)3
=(
13
)3
=127
Math 17 (UP-IMath) R and Integer Exponents Lec 2 30 / 33
![Page 114: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/114.jpg)
Laws of Exponents: Power of a Quotient Law
Let n ∈ Z, a, b ∈ R. If b 6= 0, then(a
b
)n=
an
bn.
Examples.
(x
y
)4
=x4
y4
63
183=
(618
)3
=(
13
)3
=127
Math 17 (UP-IMath) R and Integer Exponents Lec 2 30 / 33
![Page 115: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/115.jpg)
Laws of Exponents
Let n, m ∈ Z, a, b ∈ R, then
1 an · am = an+m
2 (an)m = anm
3 (ab)n = anbn
4 If a 6= 0, thenan
am= an−m
5 If b 6= 0, then(a
b
)n=
an
bn
Math 17 (UP-IMath) R and Integer Exponents Lec 2 31 / 33
![Page 116: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/116.jpg)
Example.
20x8y2z5
−5x2y7z5
=20−5· x
8
x2· y
2
y7· z
5
z5
= −4 · x8−2 · y2−7 · z5−5
= −4 · x6 · y−5 · 1
= −4x6
y5
Math 17 (UP-IMath) R and Integer Exponents Lec 2 32 / 33
![Page 117: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/117.jpg)
Example.20x8y2z5
−5x2y7z5
=20−5· x
8
x2· y
2
y7· z
5
z5
= −4 · x8−2 · y2−7 · z5−5
= −4 · x6 · y−5 · 1
= −4x6
y5
Math 17 (UP-IMath) R and Integer Exponents Lec 2 32 / 33
![Page 118: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/118.jpg)
Example.20x8y2z5
−5x2y7z5=
20−5· x
8
x2· y
2
y7· z
5
z5
= −4 · x8−2 · y2−7 · z5−5
= −4 · x6 · y−5 · 1
= −4x6
y5
Math 17 (UP-IMath) R and Integer Exponents Lec 2 32 / 33
![Page 119: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/119.jpg)
Example.20x8y2z5
−5x2y7z5=
20−5· x
8
x2· y
2
y7· z
5
z5
= −4 · x8−2 · y2−7 · z5−5
= −4 · x6 · y−5 · 1
= −4x6
y5
Math 17 (UP-IMath) R and Integer Exponents Lec 2 32 / 33
![Page 120: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/120.jpg)
Example.20x8y2z5
−5x2y7z5=
20−5· x
8
x2· y
2
y7· z
5
z5
= −4 · x8−2 · y2−7 · z5−5
= −4 · x6 · y−5 · z0
= −4x6
y5
Math 17 (UP-IMath) R and Integer Exponents Lec 2 32 / 33
![Page 121: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/121.jpg)
Example.20x8y2z5
−5x2y7z5=
20−5· x
8
x2· y
2
y7· z
5
z5
= −4 · x8−2 · y2−7 · z5−5
= −4 · x6 · y−5 · 1
= −4x6
y5
Math 17 (UP-IMath) R and Integer Exponents Lec 2 32 / 33
![Page 122: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/122.jpg)
Example.20x8y2z5
−5x2y7z5=
20−5· x
8
x2· y
2
y7· z
5
z5
= −4 · x8−2 · y2−7 · z5−5
= −4 · x6 · y−5 · 1
= −4x6
y5
Math 17 (UP-IMath) R and Integer Exponents Lec 2 32 / 33
![Page 123: A2 - Real Number System](https://reader038.fdocuments.in/reader038/viewer/2022102719/577cc58d1a28aba7119cc4c8/html5/thumbnails/123.jpg)
Exercises:
1 Let the universal set be R, X be the half-open interval[−3
2 , 5),
Y = {x | 4 ≤ |x|}, and Z be the interval (2, +∞). Find the followingsets and express them in interval notation:
1 Y ∩ Zc
2 (X ∪ Y )c
3 Xc ∩ (Z ∪X)4 X\(Y ∩ Z)
2 Simplify the following expressions:
1a−1 + b−1
(a + b)−1 2
(185 · 203x−1y2z4
159x−3y−2z−1
)−3
Math 17 (UP-IMath) R and Integer Exponents Lec 2 33 / 33