Download Chapter 10

10.1 Radical Expressions and Graphs
•
a is the positive square root of a, and  a
is the negative square root of a because
 a
2


2
 a and  a  a
• If a is a positive number that is not a perfect
square then the square root of a is irrational.
• If a is a negative number then square root of a is
not a real number.
• For any real number a: a 2  a
10.1 Radical Expressions and Graphs
• The nth root of a:
n
a is the nth root of a. It is a number whose nth
power equals a, so:
 a
n
n
a
• n is the index or order of the radical
• Example:
5
32  2 because 2  32
5
10.1 Radical Expressions and Graphs
• The
nth
root of
nth
powers:
– If n is even, then
n
– If n is odd, then
n
a
n
a
n
a
n
a
a
n
• The nth root of a negative number:
– If n is even, then the nth root is not a real number
– If n is odd, then the nth root is negative
10.1 - Graph of a Square Root
Function
f ( x) 
(0, 0)
x
10.2 Rational Exponents
•
Definition:
1
n
a n a
a
a
•

m
n

  a

m
n
1

1
n
m
n
m

 


 a

1
m
n
 
n
m
a
a
All exponent rules apply to rational exponents.
10.2 Rational Exponents
•
Tempting but incorrect simplifications:
1
a  n
a
m
m
a
n
a  n
a
1
n
a
m
n
 a
m
n
10.2 Rational Exponents
•
Examples:
2
3
4
3
5 5  5
25
25
1
4
3
4
 25
24
3
3
13
4 4
6
3
 5  5  25
 25
2
 12

1
25
1
2

1
1

25 5
10.3 Simplifying Radical
Expressions
•
Review: Expressions vs. Equations:
–
Expressions
1.
2.
3.
–
No equal sign
Simplify (don’t solve)
Cancel factors of the entire top and bottom of a fraction
Equations
1.
2.
3.
Equal sign
Solve (don’t simplify)
Get variable by itself on one side of the equation by
multiplying/adding the same thing on both sides
10.3 Simplifying Radical
Expressions
• Product rule for radicals:
n
a
n
b 
n
• Quotient rule for radicals:
n
n
a n a

b
b
ab
10.3 Simplifying Radical
Expressions
• Example:
48

3
48
 16  4
3
• Example:
3
81

3
3
3
81 3
 27  3
3
10.3 Simplifying Radical
Expressions
•
Simplified Form of a Radical:
1. All radicals that can be reduced are reduced:
9  3 and
3
a  a3 a
4
2. There are no fractions under the radical.
3. There are no radicals in the denominator
4. Exponents under the radical have no common
factor with the index of the radical
4
a2  a
2
4
a
1
2

a
10.3 Simplifying Radical Expressions
• Pythagorean Theorem: In a right triangle, with the
hypotenuse of length c and legs of lengths a and b,
it follows that c2 = a2 + b2
c
a
90
b
• Pythagorean triples (integer triples that satisfy the
Pythagorean theorem):
{3, 4, 5}, {5, 12, 13}, {8, 15, 17}
10.3 Simplifying Radical Expressions
• Distance Formula: The distance between 2
points (x1, y1) and (x2,y2) is given by the
formula (from the Pythagorean theorem):
d  x2  x1    y2  y1 
2
2
10.4 Adding and Subtracting Radical
Expressions
• We can add or subtract radicals using the
distributive property.
• Example:
5 3  2 3  (5  2) 3  7 3
10.4 Adding and Subtracting Radical
Expressions
• Like Radicals (similar to “like terms”) are terms
that have multiples of the same root of the same
number. Only like radicals can be combined.
3  2 5 cannot be combined
3  2  3 cannot be combined
3
5 3  27  5 3  9  3  5 3  9  3
 5 3 3 3 8 3
10.4 Adding and Subtracting Radical
Expressions
• Tempting but incorrect simplifications:
x  y  x y
x  y  x y
2
2
2
2
10.5 Multiplying and Dividing Radical Expressions
• Use FOIL to multiply binomials involving radical
expressions
• Example: (5  2 )( 3  6 )
5 3 5 6 
2 3
5 3 5 6 
6
5 36 6 2 3
7 36 6
12
2 6
10.5 Multiplying and Dividing
Radical Expressions
•
Examples of Rationalizing the Denominator:
3
3
3 3 3



 3
3
3
3 3
5
5
5 2
5 2
10





2
2
2
2 2
2 2
10.5 Multiplying and Dividing
Radical Expressions
•
Using special product rule with radicals:
 a  b    a  b   a 2  b2


3 1 
  
3 1 
2
3  12  3  1  2
10.5 Multiplying and Dividing
Radical Expressions
•
Using special product rule for simplifying a
radical expression:


2
2
3 1 2 3 1




2
2
3 1
3 1 3 1
3 1
2

  2
3 1
3 1

3 1
2
 
3 1
10.6 Solving Equations with
Radicals
•
•
Squaring property of equality: If both sides of an
equation are squared, the original solution(s) of
the equation still work – plus you may add some
new solutions.
Example:
x  5 has one solution (5)
x  25 has two solutions (5 and - 5)
2
10.6 Solving Equations with
Radicals
•
Solving an equation with radicals:
1. Isolate the radical (or at least one of the radicals if
there are more than one).
2. Square both sides
3. Combine like terms
4. Repeat steps 1-3 until no radicals are remaining
5. Solve the equation
6. Check all solutions with the original equation (some
may not work)
10.6 Solving Equations with Radicals
•
Example: x  3x  7  1
Add 1 to both sides: x  1 
3x  7
Square both sides: x  2 x  1  3 x  7
2
2
x
 x60
Subtract 3x + 7:
( x  3)( x  2)  0
So x = -2 and x = 3, but only x = 3 makes the
original equation equal.
10.7 Complex Numbers
•
Definition:
i   1 and i  1
2
•
•
•
Complex Number: a number of the form a + bi
where a and b are real numbers
Adding/subtracting: add (or subtract) the real
parts and the imaginary parts
Multiplying: use FOIL
10.7 Complex Numbers
•
Examples:
(2  3i )  (1  2i )  (2  1)  (3  2)i  1  5i
(4  5i )  (1  2i )  (4  1)  (5  2)i  3  3i
(2  3i )(1  2i )  2(1)  2(2i )  3i (1)  3(2)i
 2  4i  3i  6  4  7i
2
10.7 Complex Numbers
•
Complex Conjugate of a + bi:
multiplying by the conjugate:
a – bi
(2  3i)( 2  3i)  2  (3i)
2
2
 4  9i  4  (9)  13
2
•
The conjugate can be used to do division
(similar to rationalizing the denominator)
10.7 Complex Numbers
•
Dividing by a complex number:
4  5i 4  5i 2  3i


2  3i 2  3i 2  3i
8  12i  10i  15i 2

22  (3i ) 2
8  2i  15(1) 23  2i 23 2



 i
4  (9)
13
13 13