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| Properties of Real Numbers
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|
Properties |
|
Examples |
|
| 1 |
a
+ b
= b
+ a, |
|
5
+ 3 = 3 + 5, |
|
| 2 |
(a
+ b)
+ c
= a
+ (b
+ c), |
|
(2
+ 3) + 5 = 2 + (3 + 5), |
|
| 3 |
a
+ 0
= a, |
|
3
+ 0 = 3, |
|
| 4 |
a
+ (-a)
= 0, |
|
4
+ (-4)
= 0, |
|
| 5 |
a
· (b
+ c)
= a
· b
+ a
· c, |
|
2
· (4 + 7) = 2 · 4 + 2 · 7, |
|
| 6 |
a
· b
= b
· a, |
|
6
· 8 = 8
· 6 |
|
| 7 |
a
+ (-b)
= a
-
b, |
|
9
+ (-4)
= 9 -
4 = 5, |
|
| 8 |
-(a
+ b)
= -a
-
b, |
|
-
(3 + 4) = -3
-
4 = -7, |
|
| 9 |
b
-
a
= -(a
-
b), |
|
5
- 7
= - (7
-
5) = -2, |
|
| 10 |
a
· (-b)
= -a
· b, |
|
3
· (-
5) = -3
· 5 = -15, |
|
| 11 |
(-a)
· (-b)
= a
· b, |
|
(-
3)
· (-
6) = 3
· 6 = 18, |
|
| 12 |
-(-a)
= a, |
|
-
(-
7) = 7 |
|
| 13 |
a
· 0
= 0, |
|
(-11)
· 0 = 0, |
|
| 14 |
 |
|
 |
|
| 15 |
a
· 1
= 1
· a
= a, |
|
-
5
· 1 = 1
· (-
5) = -
5, |
|
| 16 |
(-1)
· a
= -a, |
|
(-1)
· 4 = -
4, |
|
| 17 |
 |
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 |
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| 18 |
 |
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| 19 |
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| 20 |
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| 21 |
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| 22 |
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| 23 |
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| 24 |
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| 25 |
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| 26 |
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| 27 |
 |
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 |
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| 28 |
 |
if |
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since |
 |
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| 29 |
 |
if |
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since |
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| Exponentiation – Powers or Indices |
Integral exponentiation – integer exponents
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|
Square or the second power
(potency)
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|
The square of a product and quotient
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|
The
rules for the manipulation of powers - Fundamental laws of
exponents (Index laws) |
|
Adding
and subtracting same powers |
|
Multiplying like bases
with exponents |
|
Dividing like bases
with exponents |
|
Zero as an exponent |
|
Power rule for exponents |
|
A product raised to an exponent |
|
A quotient raised to an exponent |
|
Negative exponents |
|
Simplifying an exponential expression |
|
Scientific notation |
|
|
Quadratic
equation
x2
=
a,
a
>
0 |
|
|
Square root |
|
Properties
of square roots |
|
Adding,
subtracting, multiplying and dividing square roots |
|
Rationalizing a denominator |
|
|
The
graph of the quadratic function
f
(x)
= x2 |
|
Translation of the source quadratic function in the direction of
the y-axis |
|
Quadratic
function of the form f
(x)
= x2 + y0 |
|
|
The principal square root function
- the inverse of the square of x (quadratic) function |
|
Definition
of the inverse function |
|
The
graph of the principal square root function |
|
|
Translation
of the principal square root function
in the direction of
the x-axis |
|
|
| Integral exponentiation – integer exponents
|
| Exponentiation,
powers, exponents (indices) |
|
|
Rules |
|
Examples |
|
| |
a
· a
· ... · a
=
an,
a
Î
Q,
n Î
N
n
factors, a
-base, n
-exponent (index) |
|
a3
= a
· a
· a,
24 =
2 ·
2 ·
2 ·
2 = 16,
(-1)
· (-1)
· (-1)
· (-1)
=
(-1)4
= 1, |
|
| |
a
= a1,
or a1
=
a, |
|
2 = 21, (-3)1
= -3, |
|
| |
a0
= 1,
an ¸
an
= 1, |
|
23
¸
23
= 23 -
3 =
20
= 1,
(ab3)0
= 1, |
|
| |
1n =
1, |
|
15
= 1,
(-1)5
= -1, |
|
| |
(-
a)2n
= a2n,
2n
-even exponents |
|
(-
5)4 = 54
= 625, |
|
| |
(-
a)2n
-1
= - a2n
-1,
2n-1
-odd exponents |
|
(-
2)3 = -
23 =
- 8, |
|
| |
(-1)2n
= 1, |
|
(-1)6
= 1, |
|
| |
(-1)2n
-1
= -1, |
|
(-1)7
= -1. |
|
|
|
| Square or the second power
|
| |
a2
=
a
· a,
12
= 1 · 1 = 1, |
|
52 =
5 ·
5 = 25,
0.42 =
0.4 ·
0.4 = 0.16, |
|
| |
(-
r)2 =
(- r)
· (-
r)
=
r2, |
|
(-
4)2 = (-
4)
· (-
4) = 16,
(-1)2 = 12
= 1. |
|
|
|
| Square of a product and quotient |
| |
(a
· b)2
=
a2
· b2, |
|
(3 ·
6)2 = 32 ·
62 = 9 ·
36 = 324, |
|
| |
a2
· b2
· c2
=
(a · b
· c)2, |
|
0.12
· 1.22
· 102 =
(0.1 ·
1.2 ·
10)2 = 1.22, |
|
| |
 |
|
 |
|
| |
(a
¸
b)2
=
a2
¸
b2, |
|
(6
¸
2)2 =
62
¸
22 = 36
¸
4 = 9, |
|
| |
a2
¸
b2
= (a
¸
b)2, |
|
642
¸
(-
16)2 =
[64
¸
(-
16)]2 = (-
4)2 = 16. |
|
|
|
| The
rules for the manipulation of exponents (or
powers) - Fundamental laws of exponents (index laws) |
| Adding
and subtracting same powers |
| Only the same powers (with same base and the exponent) can be
added or subtracted, |
|
for example a5
+ a5
+ a5
= 3a5. |
| We also say
that only like or similar terms can be added. The number that multiply the power is called |
| coefficient. We add or subtract the same powers by adding or subtracting their
coefficients. |
| Examples: |
|
a)
2a2
+ 5a
- a2
- a
= a2
+ 4a,
b) 3a4
-
7a4
+ a4
=
(3 -
7 + 1) · a4
= -3a4 |
|
|
|
| The
rules for powers (or exponents) |
|
Rules |
|
Examples |
|
|
am
· an
= am
+
n, |
|
a)
a3
· a4 =
a3
+ 4,
b) 24 · 25 = 24 + 5 =
29 = 512, |
|
|
am
¸
an =
am
-
n, |
|
a)
x5
¸
x3
= x5
-
3
= x2,
b) 0.14
¸
0.13 = 0.14
-
3 = 0.1, |
|
|
an
· bn
= (a
· b)n, |
|
a)
24 · 34 = (2
· 3)4
= 64,
b) 45 · 0.85 = (4
· 0.8)5
= 3.25, |
|
|
an
¸
bn
= (a ¸
b)n, |
|
a)
x6
¸
y6
=
(x
¸
y)6,
b) 125 ¸
35 = (12
¸
3)5 =
45, |
|
|
 |
|
 |
|
|
(am)n =
am
· n, |
|
a)
(a3)4
= a3
·
4 = a12,
b) (45)3
= 45
·
3 = 415, |
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|
| Simplifying an exponential expression |
| Use
the above rules of powers (or exponential laws) and the rules of
algebra to simplify expressions with |
| numerical and variable
bases, as show examples: |
|
|
|
|
| Scientific notation |
| A
number in scientific notation is written as the product of a
number called coefficient and a power of 10. |
| While converting to
scientific notation, the decimal point of the coefficient is
placed behind the first non-zero |
| digit. The sign of the exponent of the power
indicates, how many places the decimal point was moved to the |
| right or to the left. |
| Examples: |
|
a)
0.0000007054 = 7.054 · 10-7,
b) 5.2 · 10-4
= 0.00052 |
| |
|
a)
4507000000 = 4.507 · 109,
b) 1.04 · 105 = 104000 |
|
|
|
| Quadratic
equation
x2
=
a,
a
>
0 |
| The quadratic equation
x2
=
a
has two solutions which are
opposite numbers only if a
> 0. |
| If
a = 0 then there is a repeated solution zero, and if
a < 0 then there are no solutions in the set of real |
| numbers. |
| The solutions are also called the roots or zeros
of the quadratic equation. |
| Examples: |
a)
x2
= 16 |
|
b) (x
-
5)2
= 49 |
| |
x2
-
42 =
0 |
|
(x
-
5)2
-
72 =
0 |
| |
(x
-
4)
· (x
+
4) =
0 |
|
[(x
-
5) -
7] ·
[(x
-
5) +
7]
=
0 |
| |
x
-
4 =
0
=> x1
= 4 |
|
(x
-
5) -
7 =
0 =>
x1
= 12 |
| |
x
+
4 =
0
=> x2
= -
4 |
|
(x
-
5) +
7 =
0
=> x2
= -2 |
|
|
| Square
root |
| The
principal square root of a
nonnegative real number
a,
denoted |
 |
represents the
nonnegative real |
|
| number
x
whose
square (the result of multiplying the number by itself) is
a,
that is |
|
|
|
The number a
under the root sign is called the radicand.
|
|
As the square root is defined only
when its radicand is a
nonnegative number therefore,
|
| for
example, the expression |
 |
makes sense only if x
-
5
> 0, that is,
only if x
> 5. |
|
|
Since the
positive square root of a positive real number is called the
principal square root this is why the
|
| square root of say 16, is
taken as + 4 despite the fact that the square of
- 4
is also
16.
|
|
That is, by the square
root of a,
where a is
a nonnegative number, we should mean, the positive square root
|
| of
a.
|
| The
reason for taking the principal square root as the
value of a square root of a positive real number, lies
in |
| the definition of a function f
which requires that for each x
in the domain there is at most one pair (x,
y)
in the |
| codomain (where a set of ordered pairs (x,
y)
represents the graph of the function f
). |
|
| As
a2
> 0 whether
a
> 0 or
a
< 0 we write
|
|
|
| Examples: |
|
 |
| |
|
 |
|
|
| Properties
of square roots |
|
|
|
| Adding,
subtracting, multiplying and dividing square roots |
|
|
|
| Rationalizing
a denominator |
| Rationalizing a denominator is a method for changing an irrational denominator into a rational one. |
|
|
| To
rationalize a denominator or numerator of the form |
 |
multiply both numerator |
|
| and denominator by a
conjugate, where |
 |
are conjugates of each other. |
|
|
| The
graph of the quadratic function
f
(x)
= x2 |
| A function that to every real number associates its square is called
a quadratic function and is denoted |
| f
(x)
= x2,
x
Î
R.
The point P(x,
x2)
lies on the graph of a quadratic function called a parabola. |
|
|
| For
x
=
0 function
f
(x)
= x2
has minimal value f
(0)
=
02 =
0.
This point is called the turning point or the
|
| vertex
of the parabola. |
| The
curve is symmetrical about the y-axis and has its vertex
V(0,
0) at the origin. |
| The curve is decreasing for
x < 0 and is increasing for
x
> 0. |
| If
y
=
f
(x),
then y
=
-
f
(x)
is its reflection about the x-axis. |
| Therefore,
the graph of the quadratic f
(x)
= -x2
has its maximum at the vertex. |
| The curve is
increasing for
x < 0 and is decreasing
for x > 0. |
|
|
Translation
(or shift) of the source quadratic function in the direction of
the y-axis, |
| quadratic
function of the form f
(x)
= x2 + y0 |
| Translating
the graph of the source quadratic function vertically by y0,
the vertex of the function moves to the |
| point V(0,
y0 ). |
| The
translation or shift is in the positive direction of the y-axis
(upward) if y0
> 0, in the negative direction |
| (downward) if
y0
< 0. |
 |
|
| Points
where a graph crosses or touches the x-axis
are called x-intercepts,
roots or zeros. At the x-intercept
|
| y
= 0. |
| To
find the zeros of the quadratic function, set the function equal
to zero, f
(x)
= 0,
and solve for x. |
| That
is, solve the equation x2 + y0
= 0, |
 |
|
|
|
| Quadratic
equation
x2
=
a,
a
>
0 |
| If
a
> 0 then the quadratic equation
x2
=
a
has two solutions, |
 |
|
| If a
= 0 then the
equation has zero as the double root, and if a
< 0 then the equation has no real roots. |
|
|
|
| The principal square root function
- the inverse of the square of x (quadratic) function |
| Definition
of the inverse function |
| The
inverse function is a function, usually written f
-1,
whose domain and range are respectively the range |
| and
domain of a given function f,
that is |
| |
f
-1(x)
= y if
and only if f
(y)
= x, |
|
|
| or it is
the function whose composition with the given function
is the identity function, i.e., |
 |
|
|
|
|
|
|
|
| In
order that the inverse should have a unique value for each
argument, and so be properly a function, |
| the
extraction of positive square roots
is the inverse of squaring, since |
|
|
| however,
without the restriction to positive values, the square root
function on the domain of real numbers |
| does not have an inverse. |
|
| The
graph of the principal square root function |
| The
graph of the inverse function is the reflection about the line y =
x
of the graph of a given function. |
| A
function f has an inverse if and only if when its graph is
reflected about the line y =
x, the result is the |
| graph of a
function that passes the vertical line test. |
| A
relation is a function if there are no vertical lines that
intersect the graph at more than one point. |
 |
|
|
Translation of the principal square root function
in the direction of
the x-axis |
| Horizontal
line test |
| A
function f
has an inverse if no horizontal line intersects the graph of
f
more than once. |
| If
any horizontal line intersects the graph of f
more than once, then f
does not have an inverse. |
| A
mapping associating a unique member of the codomain with every
member of the domain of a function is |
| called one to one
correspondence. |
| A
function f
is one-to-one if and only if f
has an inverse. |
| Given
f
(x)
= x2 + y0
and,
since f
[f
-1(x)]
= x |
|
therefore, f
[f
-1(x)]
= [f
-1(x)]2
+ x0
= x, or
[f
-1(x)]2
= x -
x0 |
| then |
 |
|
|
| To find the
y-intercept,
set x
= 0 and solve for y, that is, |
 |
|
|
|
| The
graph of
translated principal square root function
in the direction of
the x-axis |
 |
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| Beginning
Algebra Contents |
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|
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
