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Chapter 7
Square Root
The number b is a square root of a if b2 = a
Example 100 = 102 = 10
radical sign
Under radical sign the expression is called radicand
Expression containing a radical sign is called a radical
expression.
Radical expressions are 6, 5 + x + 1 , and 3x
2x - 1
Cube Root
The number b is a cube root of a if b3 = a
Example – Find the cube root of 27
3
27
=
3 33
=3
Estimating a cellular phone
transmission distance
R
The circular area A is covered by one transmission tower is A =
2
The total area covered by 10 towers are 10
R
R2
, which must
equal to 50 square miles
Now solve R
R = 1.26, Each tower must broadcast with a minimum radius of
approximately 1.26 miles
Expression
 For every real number
If n is an integer greater than 1, then a1 n = n a
Note : If a < 0 and n is an even positive integer, then
a1 n is not a real number.
 If m and n are positive integer with m/n in lowest
terms, then
a mn = n a m = ( n a ) m
Note : If a < 0 and n is an even integer, then a m n
is not a real number.
If m and n are positive integer with m/n in lowest terms,
then
a - m n = 1/ a m n
a =0
Properties of Exponent
Let p and q be rational numbers. For all real numbers a
and b for which the expressions are real numbers the
following properties hold.
1
2
3
4
5
6
7
a p . a q = a p+q
a - p = 1/ a p
a/b -p = b a p
a p = a p-q
aq
a p q = a pq
ab p= a p b p
a
b
p =
ap
bp
Product rule
Negative exponents
Negative exponents for quotients
Quotient rule for exponents
Power rule for exponents
Power rule for products
Power rule for products
Power rule for quotients
7.2 Simplifying Radical Expressions
Let a and b are real numbers where a and b are both
defined. = n a , n b
Product rule for radical expression (Pg – 509)
n
n
n
ab
a . b=
Quotient rule for radical expression where b = 0 (Pg 512)
n
a
=
b
n
n
a
b
Square Root Property
Let k be a nonnegative number. Then the
solutions to the equation.
x2 = k
are x = + k. If k < 0. Then this equation has no
real solutions.
Using Graphing Calculator
[ 5, 13, 1] by [0, 100, 10]
To find cube root technologically
7.3 Operations on Radical Expressions
Addition
10 11 + 4
11
= (10 + 4) 11
= 14
11
53 6  3 6  (5  1)3 6  63 6
Subtraction
10
11 -
4
11=
(10 - 4)
11
=6
11
53 6  3 6  (5  1)3 6  43 6
Rationalize the denominator (Pg 484)
Using Graphing Calculator
Y1 = x2
[ -6, 6, 1] by [-4, 4, 1]
Pg -522
Rationalizing Denominators having square roots
1
3
3
3
=
3
3
7.6 Complex Numbers
Pg 556
x2 +1=0
x 2 = -1
x =+ - 1
Square root property
Now we define a number called the imaginary unit, denoted by i
Properties of the imaginary unit i
i=
-1
A complex number can be written in standard form, as a + bi, where a and b
are real numbers. The real part is a and imaginary part is b
Pg 513
a + ib
Complex Number
-3 + 2i
Real part a
Imaginary Part b
-3
2
5 -3i -1 + 7i - 5 – 2i 4 + 6i
5
-1
-5
4
-3
7
-2
6
Complex numbers contains the set of
real numbers
Complex numbers
a +bi
a and b real
Real numbers
a +bi
Rational Numbers
-3, 2/3, 0 and –1/2
Imaginary Numbers
b=0
a +bi
Irrational numbers
3 And
- 11
b =0
Sum or Difference of Complex
Numbers
Let a + bi and c + di be two complex numbers.
Then
Sum
( a + bi ) + (c +di) = (a + c) + (b + d)i
Difference
(a + bi) – (c + di) = (a - c) + (b – d)i