Chapter 7
Square RootThe number b is a square root of a if b2 = a
Example 100 = 102 = 10
radical sign
Under radical sign the expression is called radicandExpression 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 = R2 2
The total area covered by 10 towers are 10 R , which must equal to 50 square milesNow solve RR = 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 m n = 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 ExponentLet p and q be rational numbers. For all real numbers a and b for which the expressions are real numbers the following properties hold.
a p . a q = a p + q Product rule
a - p = 1/ a p Negative exponents a/b -p = b a p Negative exponents for quotients
a p = a p-q Quotient rule for exponents
a q
a p q = a pq Power rule for exponents ab p= a p b p Power rule for products
a p = a p Power rule for products
b b p Power rule for quotients
12
34
56
7
7.2 Simplifying Radical Expressions
Let a and b are real numbers where a and b are both defined.
Product rule for radical expression (Pg – 509)n a n b
=
n ab . =Quotient rule for radical expression where b = 0 (Pg 512)
=n
b
a n a
n b
n bn a ,
Square Root Property
Let k be a nonnegative number. Then the
solutions to the equation.
x2 = kare 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 + 4 = (10 + 4) = 14
Subtraction
11 11 11 11
10 - 4 = (10 - 4) = 611 11 11 11
3333 666)15(665
3333 646)15(665
Rationalize the denominator (Pg 484)
Using Graphing Calculator
Y1 = x2
[ -6, 6, 1] by [-4, 4, 1]
Pg -522
Rationalizing Denominators having square roots
3
1
3
3 = 3
3
7.6 Complex NumbersPg 556
x 2 + 1 = 0x 2 = -1x =+
- 1 Square root property
- 1i =
Now we define a number called the imaginary unit, denoted by i
Properties of the imaginary unit i
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
Complex Number -3 + 2i 5 -3i -1 + 7i - 5 – 2i 4 + 6i
Real part a - 3 5 -1 -5 4
Imaginary Part b 2 -3 7 -2 6
a + ib
Complex numbers contains the set of real numbers
Complex numbers
a +bi a and b real
Real numbersa +bi b=0
Imaginary Numbersa +bi b =0
Rational Numbers-3, 2/3, 0 and –1/2
Irrational numbers 3 And - 11
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
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