Download Elementary Algebra Test #8 Review

Elementary Algebra
Test #8 Review
Page 1 of 7
Section 8.1: Evaluating Roots
ROOTS “UNDO” POWERS!
Square Roots:
If a is a positive real number, then
a is the principal square root of a, and
- a is the negative square root of a.
For nonnegative a,
a a =a
and (- a )  (- a ) = a.
Application of Square Roots: the Pythagorean Theorem
Pythagorean Theorem:
a 2  b2  c2
Higher order roots:
Definition: The principal nth root of a number
n
a , where n is an integer greater than or equal to 2, computes to a number b such that if
n
a  b , then b n  a .
If n is an even number bigger than 2, then a and b must be positive.
If n is an odd number, then a and b can be any real number.
Elementary Algebra
Test #8 Review
Page 2 of 7
n
2
3
4
5
6
7
8
9
10
2^n
4
8
16
32
64
128
256
512
1,024
3^n
4^n
5^n
6^n
7^n
8^n
9^n
10^n
9
16
25
36
49
64
81
100
27
64
125
216
343
512
729
1,000
81
256
625
1,296
2,401
4,096
6,561
10,000
243
1,024
3,125
7,776
16,807
32,768
59,049
100,000
729
4,096
15,625
46,656
117,649
262,144
531,441 1,000,000
2,187
16,384
78,125
279,936
823,543 2,097,152 4,782,969
6,561
65,536
390,625 1,679,616 5,764,801
19,683
262,144 1,953,125
59,049 1,048,576 9,765,625
n
2
3
4
5
6
7
8
9
10
(-2)^n
4
-8
16
-32
64
-128
256
-512
1,024
(-3)^n
(-4)^n
(-5)^n
(-6)^n
(-7)^n
(-8)^n
(-9)^n
(-10)^n
9
16
25
36
49
64
81
100
-27
-64
-125
-216
-343
-512
-729
-1,000
81
256
625
1,296
2,401
4,096
6,561
10,000
-243
-1,024
-3,125
-7,776
-16,807
-32,768
-59,049 -100,000
729
4,096
15,625
46,656
117,649
262,144
531,441 1,000,000
-2,187
-16,384
-78,125 -279,936 -823,543 -2,097,152 -4,782,969
6,561
65,536
390,625 1,679,616 5,764,801
-19,683 -262,144 -1,953,125
59,049 1,048,576 9,765,625
Section 8.2: Multiplying, Dividing, and Simplifying Radicals
Product Rule for Square Roots: For nonnegative real numbers a and b,
a  b  ab (the product of two square roots is the square root of the products)

 and ab  a  b (vice-versa; the square root of a product equals the product of the square roots).
This second equation is very useful for simplifying radicals, if you can think of the number under the square
root as a product of a perfect square and another number.
Criteria for a Simplified Radical Expression:

There are as few radicals in the expression as possible.

The radicands are as small as possible.
Quotient Rule for Square Roots: For nonnegative real numbers a and b, and b  0,
a
a

(the square root of a quotient is the quotient of the square roots)

b
b

And
a
a
(the quotient of two square roots is the square root of the quotient)

b
b
Elementary Algebra
Test #8 Review
Page 3 of 7
The Square Root of a Square: For any number a,


a 2  a (the square root of a square is the absolute value of squared value)
Notice:
 4
2
 16  4  4
Product Rule for Radicals: For all real numbers for which the indicated roots exist,
 n a  n b  n ab

n
n
a na

b
b
Section 8.3: Adding and Subtracting Radicals
Big Idea: Radicals can only be added or subtracted when they are like radicals. Like radicals have the same
order and same radicand. If the radicals are not similar, then they can not be combined by addition or
subtraction.
 Example of similar radicals that can be added:
43 7  53 7  93 7
 Example of radicals that are not similar because of different orders and thus can not be added:
3
757
 Example of radicals that are not similar because of different radicands and thus can not be added:
11  17
 Radicals that do not look similar but can be shown to be similar when they are simplified first:
8  2  42  2  2 2  2  3 2
Criteria for a Simplified Radical Expression:

There are as few radicals in the expression as possible.

The radicands are as small as possible.
Section 8.4: Rationalizing the Denominator
Criteria for a Simplified Radical Expression:

There are as few radicals in the expression as possible.

The radicands are as small as possible.

The radicand has no fractions.

No denominator contains a radical.
Section 8.5: More Simplifying and Operations with Radicals
Criteria for a Simplified Radical Expression:

There are as few radicals in the expression as possible.

The radicands are as small as possible.

The radicand has no fractions.

No denominator contains a radical.
Elementary Algebra
Test #8 Review
Page 4 of 7
To multiply radical expressions with terms, use the distributive property.
Examples:


3
5 42 5  4 5 2 5 5


6  5 2 6  3 5  3 6  2 6  3 6 3 5  5  2 6  5 3 5
 4 5  2 10
 6 6 6 9 6 5 2 5 6 3 5 5
 20  4 5
 6  6  9 30  2 30  3  5
 36  7 30  15
 21  7 30
Rationalizing a denominator with two (square root) terms:
Multiply top and bottom by the conjugate of the denominator so that the denominator becomes a difference of
squares.
Section 8.6: Solving Equations with Radicals
Big Idea: To solve an equation with a radical, isolate the radical, then raise both sides to an appropriate power
to get rid of the radical.
Section 8.7: Using Rational Numbers as Exponents
We have seen that a square root “undoes” or “cancels” a power of 2:
25  52  5
But notice what raising a power of 2 to the power of ½ does:
25 2   52   52 2  51  5
1
1
1
2
So, a square root produces the same result as a power of ½.
The same comparison for cube roots and a power of
3
1
3
:
8  3 23  2
8 3   23   23 3  21  2
1
1
3
1
Thus, taking the nth root of a number is the same thing as raising it to a power of
n
a a
1
n
Other results based on this idea and the rules of exponents:
n
a m  a m n  a n  a
1
m
1
n m
   a
 a
1
m
n
n
m
1
n
:
Elementary Algebra
Test #8 Review
Page 5 of 7
Section 9.1: Solving Quadratic Equations by the Square Root Property
The Square Root Property
If k is a positive number and x 2  k then x  k or x   k . The solution set is  k , k , which can be




written as  k .
To Solve a Quadratic Equation Using the Square Root Property:

Isolate the square.

Use the square root property.
Section 9.2: Solving Quadratic Equations by Completing the Square
To solve quadratic equations that can’t be factored, we manipulate the equation so that it becomes the square of
a binomial plus a constant. This manipulation involves taking the first two terms, and finding out what we have
to add to them to make a perfect square trinomial, which can be replaced with the square of a binomial.
So, to solve x 2  2 x  1  0 , the first two terms are identical to the first two terms of the perfect square trinomial
2
x 2  2 x  1 , which comes from the square of the binomial x + 1:  x  1  x 2  2 x  1 . So, here is what we do:
x2  2x 1  0
x2  2x  1
x2  2x  1  1  1
 x  1
2
2
 x  1
2
 2
x 1   2
x  1  2
x  1  2
OR
x  1  2
x  0.414
OR
x  2.414
To Solve a Quadratic Equation by Completing the Square :
(i.e, writing a quadratic trinomial as a perfect square trinomial plus a constant)
 Get the constant term on the right hand side of the equation.
i.e., if x 2  bx  c  0 , then write the equation as x 2  bx  c
 Make sure the coefficient of the square term is 1.
 Identify the coefficient of the linear term; multiply it by ½ and square the result.

1 
i.e., Find the number b in x 2  bx  c and compute  b 
2 
Add that number to both sides of the equation.
2
1 
1 
i.e., x 2  bx   b   c   b 
2 
2 
2
2
Elementary Algebra
Test #8 Review
Page 6 of 7

Write the resulting perfect square trinomial as the square of the binomial .

1 

1 
i.e.,  x  b   c   b 
2 

2 
Use the square root property to solve the equation.
2
2
Section 9.3: Solving Quadratic Equations by the Quadratic Formula
Quadratic Formula
The solutions of the quadratic formula ax 2  bx  c  0 (a  0) are:
b  b2  4ac
b  b2  4ac
and
x
x
2a
2a
2
b  b  4ac
or, in compact form, x 
.
2a
Section 9.4: Complex Numbers
Definition: The b
The imaginary unit, denoted by i, is the number whose square root is -1. That is:
i2 = -1
OR
i  1
Thus, for any positive real number b, b  i b .
Definition: Complex Numbers
A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit.
If a = 0 and b  0 the number bi is called a pure imaginary number.
The standard form for writing a complex number is the form a + bi.
A pure imaginary number is a complex number of the form bi.
Note: under this definition, the real numbers are a subset of the complex numbers.
Evaluating Square Roots of Real Numbers
If N is a positive real number, then we define the principal square root of –N, denoted as
 N  Ni ,
where i is the imaginary unit.
 N , as
Elementary Algebra
Test #8 Review
Page 7 of 7
THE ARITHMETIC OF COMPLEX NUMBERS
Note: Adding, subtracting, multiplying, or dividing complex numbers results in an answer that is also a
complex number.
To add complex numbers, add “like terms”:
 a  bi    c  di    a  c   b  d  i
To subtract complex numbers, subtract “like terms”:
 a  bi    c  di    a  c   b  d  i
To multiply complex numbers, use the distributive property, then combine “like terms” (like using FOIL):
Definition: conjugate of a complex number
The conjugate of a complex number a + bi is a – bi.
Product of conjugates:
 a  bi  a  bi   a2  b2
To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator,
then multiply and simplify. Note: the denominator will always multiply out to a difference of squares, which
will be a real number.
Example:
4  3i 4  3i 1  2i


1  2i 1  2i 1  2i
 4  3i 1  2i 

1  2i 1  2i 

4  8i  3i  6i 2
12   2i 
2
4  11i  6
1  4i 2
2  11i

1 4
2 11i


5
5

Section 9.5: More on Graphing Quadratic Equations; Quadratic Functions
Big Idea: Defining a new set of numbers that includes the square root of negative numbers lets us write down
answers to many types of equations that we couldn’t otherwise.
Big Skill: You should be able to accurately graph quadratic functions.