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Algebra 2 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 5-1 Monomials
Lesson 5-2 Polynomials
Lesson 5-3 Dividing Polynomials
Lesson 5-4 Factoring Polynomials
Lesson 5-5 Roots of Real Numbers
Lesson 5-6 Radical Expressions
Lesson 5-7 Rational Exponents
Lesson 5-8 Radical Equations and Inequalities
Lesson 5-9 Complex Numbers
Example 1 Simplify Expressions with Multiplication
Example 2 Simplify Expressions with Division
Example 3 Simplify Expressions with Powers
Example 4 Simplify Expressions Using
Several Properties
Example 5 Express Numbers in Scientific Notation
Example 6 Multiply Numbers in Scientific Notation
Example 7 Divide Numbers in Scientific Notation
Multiplying Monomials:
• When multiplying monomials you must
ADD exponents.
• Example: 2x3  3x5  2x x x  3x x x x x
 6x8
Definition of exponents
Commutative Property
Answer:
Definition of exponents
Answer:
Try These
Multiply the following monomials.
1. a2 • a6
2. 3x2 • 7x4
3. (-3b3c)(7b2c2)
4. 2x2(6y3)(2x2y)
Try These
Multiply the following monomials.
1. a2 • a6
a8
2. 3x2 • 7x4
21x6
3. (-3b3c)(4b2c2)
4. 2x2(6y3)(2x2y)
-12b5c3
24x4y4
Dividing Monomials:
• When dividing monomials you must
SUBTRACT exponents.
7
6x
• Example:

5
2
x
•
• Cancel x’s
6 xxxxxxx
2 xxxxx
 3xx  3x2
Subtract exponents.
Remember that a simplified
expression cannot contain
negative exponents.
1
1
Answer:
1
1
Simplify.
Answer:
Try These
Divide the following monomials.
1.
3.
6
2.
5 3 3
4.
2
a n
5
an
3a b c
3 7
9a b c
y z
2 5
y z
5 7
3
2
5
2c d (3c d )
4 2
30c d
Try These
Divide the following monomials.
1.
3.
2
2.
6
a n
5
an
5 3 3
3a b c
3 7
9a b c
an
4.
y z
2 5
y z
5 7
3
-y3z2
2
5
2c d (3c d )
4 2
30c d
Power to a Power:
• When raising a power to a power you must
MULTIPLY exponents.
• Example: (x3)5  This means 5 groups of
(x3).  (x3)  (x3)  (x3)  (x3)  (x3)
• (xxx)(xxx)(xxx)(xxx)(xxx)
• x15
Product to a Power:
• When raising a product to a power you
raise every number/variable to that power.
• Example: (2x2y3) 6  (2x2y3)  (2x2y3) 
(2x2y3)  (2x2y3)  (2x2y3)  (2x2y3)
which can be written as:
(2xxyyy) (2xxyyy) (2xxyyy) (2xxyyy)
(2xxyyy) (2xxyyy)
 64x12y18
Quotient to a Power
• When raising a quotient to a power you
raise the numerator & denominator to that
power.
3
x
x
 x
   3 
125
5
5
3
3
Power of a power
Answer:
Answer:
Power of a power
Power of a quotient
Power of a product
Answer:
Negative exponent
Power of a quotient
Answer:
Simplify each expression.
a.
Answer:
b.
Answer:
c.
Answer:
d.
Answer:
Try These
Simplify each monomial.
1. (n4)4
2. (2x)4
3. (-2r2s) 3 (3rs2)
4.
6x2 y4 3
( 4 3)
3x y
Try These
Simplify each monomial.
1. (n4)4
2. (2x)4
n16
3. (-2r2s) 3 (3rs2)
-24r7n5
16x4
4.
6x2 y4 3
( 4 3)
3x y
Negative Exponents
• To make a negative exponent positive,
move the number/variable that is being
raised to that exponent from the numerator
to the denominator or vice versa.
• Example:
x-3

1
x3
Try These
Simplify each monomial.
2
1. 15 x y
4
5 x 2 y 6
3. (a3b3)(ab)-2
2. 15 x 2 y 4
5 x 2 y 6
4.
xy  2
( 2)
y
Try These
Simplify each monomial.
1.
28 x 4
6 x 2
3. (a3b3)(ab)-2
2. 15 x 2 y 4
5 x 2 y 6
4.
xy  2
( 2)
y
Method 1
Raise the numerator and the denominator to
the fifth power before simplifying.
Answer:
Method 2 Simplify the fraction before raising to the fifth
power.
Answer:
Answer:
Express 4,560,000 in scientific notation.
4,560,000
Answer:
Write 1,000,000
as a power of 10.
Express 0.000092 in scientific notation.
Answer:
Use a negative exponent.
Express each number in scientific notation.
a. 52,000
Answer:
b. 0.00012
Answer:
Express the result in
scientific notation.
Associative and
Commutative Properties
Answer:
Express the result in
scientific notation.
Associative and
Commutative Properties
Answer:
Evaluate. Express the result in scientific notation.
a.
Answer:
b.
Answer:
Biology There are about
red blood cells in one
milliliter of blood. A certain blood sample contains
red blood cells. About how many milliliters
of blood are in the sample?
Divide the number of red blood cells in the sample by the
number of red blood cells in 1 milliliter of blood.
 number of red blood cells in sample
 number of red blood cells in 1 milliliter
Answer: There are about 1.66 milliliters of blood
in the sample.
Biology A petri dish started
with
germs in it.
A half hour later, there are
How many times as
great is the amount a half
hour later?
Answer:
Assignment:
Page 226 #26, 32, 36, 40
Example 1 Degree of a Polynomial
Example 2 Subtract and Simplify
Example 3 Multiply and Simplify
Example 4 Multiply Two Binomials
Example 5 Multiply Polynomials
Polynomials
• Polynomial: The sum of terms such as 5x,
3x2, 4xy, 5
• Polynomial Terms have variable and whole
number exponents. There are no square
roots of exponents, no fractional powers,
and no variables in the denominators.
Polynomials
6x-2
Not a
Has a negative
polynomial term exponent
1/x2
Not a
Has variable in
polynomial term the
denominator.
Not a
Has variable in
polynomial term radical.
4x2
Is a polynomial
term
Determine whether
is a polynomial. If it
is a polynomial, state the degree of the polynomial.
Answer: This expression is not a polynomial
because
is not a monomial.
Determine whether
is a polynomial.
If it is a polynomial, state the degree of the polynomial.
Answer: This expression is a polynomial because each
term is a monomial. The degree of the first term
is 5 and the degree of the second term is 2 + 7
or 9. The degree of the polynomial is 9.
Determine whether each expression is a polynomial. If
it is a polynomial, state the degree of the polynomial.
a.
Answer: yes, 5
b.
Answer: no
Adding Polynomials
•
1.
2.
3.
•
•
•
Add: (2x2 - 4) + (x2 + 3x - 3)
Remove parentheses.
Identify like terms.
Add the like terms.
(2x2 - 4) + (x2 + 3x - 3)
= 2x2 - 4 + x2 + 3x - 3
= 3x2 + 3x - 7
Subtracting Polynomials
•
1.
2.
3.
4.
Subtract: (2x2 - 4) - (x2 + 3x - 3)
Remove parentheses.
Change the signs of ALL of the terms being
subtracted.
Change the subtraction sign to addition.
Follow the rules for adding signed numbers.
•
•
•
•
•
(2x2 - 4) - (x2 + 3x - 3)
(Change the signs of terms being subtracted)
= (2x2 - 4) + (-x2 - 3x + 3)
= 2x2 - 4 + -x2 - 3x + 3
= x2 - 3x - 1
Simplify
Distribute the –1.
Group like terms.
Answer:
Combine like terms.
Simplify
Answer:
Try These
Add or subtract as indicated.
1. (3x2 – x – 2) + (x2 + 4x – 9)
2. (5y + 3y2) + (– 8y – 6y2)
3. (9r2 + 6r + 16) – (8r2 + 7r + 10)
4. (10x2 – 3xy + 4y2) – (3x2 + 5xy)
Try These
Add or subtract as indicated.
1. (3x2 – x – 2) + (x2 + 4x – 9)
2. (5y +
3y2)
+ (– 8y –
6y2)
4x2 + 3x - 11
-3y – 3y2
3. (9r2 + 6r + 16) – (8r2 + 7r + 10) r2 – r + 6
4. (10x2 – 3xy + 4y2) – (3x2 + 5xy) 7x2 -8xy + 4y2
Multiplying Polynomials
• Simply multiply each term from the first
polynomial by each term of the second
polynomial.
• Example:
• (x + 3)(x² + 2x + 4)
• = x³ + 2x² + 4x + 3x² + 6x + 12
• = x³ + 2x² + 3x² + 4x + 6x + 12
• = x³ + 5x² + 10x + 12
Distributive Property
Answer:
Multiply the monomials.
Answer:
+
+
+
First terms Outer terms Inner terms Last terms
Answer:
Multiply monomials
and add like terms.
Answer:
Distributive Property
Distributive Property
Multiply monomials.
Answer:
Combine like terms.
Answer:
Try These
Multiply the polynomials.
1. 4b(cb – zd)
2. 2xy(3xy3 – 4xy + 2y4)
3. (3x + 8)(2x + 6)
4. (x – 3y)2
5. (x2 + xy + y2)(x – y)
Try These
Multiply the polynomials.
1. 4b(cb – zd)
4b2c – 4bdz
2. 2xy(3xy3 – 4xy + 2y4)
6x2y4 – 8x2y2 + 4xy5
3. (3x + 8)(2x + 6)
6x2 + 34x + 48
4. (x – 3y)2
x2 – 6xy + 9y2
5. (x2 + xy + y2)(x – y)
x 3 – y3
Assignment:
Page 231-232 #25, 26, 30,
42
Example 1 GCF
Example 2 Grouping
Example 3 Two or Three Terms
Example 4 Quotient of Two Trinomials
Factoring Lesson #1
• Greatest Common Factor
• Polynomials in the form x2 + bx + c
Greatest Common Factor
• The first thing you should always do when factoring is to
take out a common factor. This is the simplest technique
of factoring, but it is important even when you learn
fancier techniques, because you will make your later
work much easier if you always look for common factors
first. Taking out common factors is using the distributive
property backwards. The distributive property says:
a(b+c)=ab+ac
• The idea behind taking out a common factor is to look for
something that all terms have “in common.” Look at thr
right side of the above equation. There is a common
factor, a.
Greatest Common Factor
• A good trick for finding
the greatest common
factor to factor
polynomials is to find the
greatest common factor
of the numbers and the
smaller power of the
variable, so here the
greatest common factor
of the numbers is 4 and
the smallest power of x is
3, so we can take out 4x3
as a common factor.
Example:
The polynomial:
4x5+12x4-8x3
Can be factored into:
4x3(x2+3x-8)
Example 1:
Factor the polynomial:
2x2 + 6x4
Now Check your work:
2x2 (1 + 3x2)
by taking out a common
factor.
Multiply back together:
2x2 + 6x4
Solution: Choose the
common factor. 2x2.
2x2 (1 + ___ )
2x2 (1 + 3x2)
Example 2:
Factor the polynomial:
15x2y – 10xy2
Now Check your work:
5xy (3x – 2y)
by taking out a common
factor.
Multiply back together:
15x2y
Solution: Choose the
common factor: 5xy.
5xy (3x – ___ )
5xy (3x – 2y)
–
10xy2
Example 3:
Factor the polynomial:
16a3b 5 – 24a2b4 – 8a4b7c
Now Check your work:
8a2b4 (2ab – 3 – a2b3c )
by taking out a common
factor.
Multiply back together:
16a3b 5 – 24a2b4 – 8a4b7c
Solution: Choose the
common factor: 8a2b4.
8a2b4 (2ab – ___ – ___ )
8a2b4 (2ab – 3 – ___ )
8a2b4 (2ab – 3 – a2b3c )
Now Try These:
Factor the following polynomials and
check your work.
a. 6x2y3 + 8x2y5
Solution: 2x2y3 (3 + 4y2)
b. 12a4b2c3 – 18ab2c4 + 24a5b3c4
Solution: 6ab2c3 (2a3 – 3c + 4a4bc)
Factoring Polynomials in the form
x2 + bx + c (General Quadratics)
Examples of these “General Quadratics”
are:
a. x2 + 7x + 10
b. x2 + 13x - 30
c. x2 - 8x + 15
d. x2 - 8x - 20
Rules for Factoring General
Quadratics
If the constant term is
positive:
-
-
Choose factors of the
constant term whose
SUM is the middle
term.
Use the same signs –
the sign of the middle
term.
• Example:
x2 + 10x + 16
(x
)( x
)
Choose factors of 16 whose sum is 10
(8 and 2)
(x
8
)( x
2 )
Use the same signs – sign of middle term
(+)
( x + 8 )( x + 2 )
Rules for Factoring General
Quadratics
If the constant term is
negative:
-
-
Choose factors of the
constant term whose
DIFFERENCE is the
middle term.
Use different signs –
the larger factor gets
the sign of the middle
term.
• Example:
x2 - 2x - 24
(x
)( x
)
Choose factors of 24 whose difference
is 2 (6 and 4)
(x
6
)( x
4 )
Use different signs – the six gets the sign
of middle term (-)
( x - 6 )( x + 4 )
Now Try These:
Factor the following polynomials and
check your work.
a. x2 + 7x + 10
b. x2 + 13x - 30
Answer: (x + 5)(x + 2)
Answer: (x + 15)(x – 2)
c. x2 - 8x + 15
d. x2 - 8x – 20
Answer: (x - 5)(x – 3)
Answer: (x – 10)(x + 2)
Part I
Warm Up – Section 5-4 #1
Factor and check.
1. 10x2y + 15x3y2
2. 16a2b4c5 + 48a3bc2 – 12ab4c3
3. y2 + 11y + 24
4. y2 - 15y + 36
5. y2 + 7y - 30
6. y2 - 4y - 45
Factoring Lesson #2
• Polynomials in the form ax2 + bx + c
Factoring Polynomials in the form
ax2 + bx + c (Trial and Error)
Examples of these “Trial and Error”
Quadratics are:
a. 4x2 - 8x - 45
b. 12x2 + 13x - 14
c. 15x2 - 26x + 7
d. 25x2 +15x + 2
Rules for Factoring General
Quadratics in the form ax2 + bx + c
-
-
List all of the possible
factors of the first term
and the last term.
Choose the
combination that will
allow you to get the
correct middle term.
• Example:
4x2 - 24x + 35
4x1
2x2
5x7
35 x 1
Choose factors whose combination will
give you the middle term (-24x). You
may have to try different combinations
before finding the one that works.
( 2x
)( 2x
)
Use the same signs – sign of middle term
(-)
-
Check your work!!!
( 2x - 7 )( 2x - 5 )
Now Try These:
Factor the following polynomials and
check your work.
a. 2x2 + 7x + 6
b. 3x2 + 10x + 3
Answer: (2x + 3)(x + 2)
Answer: (3x + 1)(x + 3)
c. 15x2 - 38x + 7
d. 10x2 - 3x – 27
Answer: (5x - 1)(3x – 7)
Answer: (5x – 9)(2x + 3)
Part II
Warm Up – Section 5-4 #2
Factor using trial and error or the junk method
and check your work.
1. 2x2 + 11x + 14
2. 14y2 – 19y – 3
3. 3a2 – 22a + 24
Factor.
4. 25r2s4t + 100rs2t3
5. x2 – 11x + 24
6. x2 + 2x – 35
Factoring Lesson #3
• Difference of two perfect squares x2 - y2
• Factoring by grouping
• Factoring Completely
Factoring the difference of two
perfect squares
Examples of these polynomials are:
a. 4x2 – 9
b. 16x2 – 36
c. x2 – 4
d. 25x2 – 16y2
Rules for Factoring the difference
of two perfect squares
-
-
-
The square root of the
first term becomes the
first term of each
binomial.
Example:
x2 - 64
The square root of the
second term becomes
the second term of
each binomial.
(x
Use different signs.
Since the square root of x2 is x, x
is the first term of each binomial.
)( x
)
Since the square root of 64 is 8, 8
is the second term of each
binomial.
(x
8 )( x
8 )
Use different signs.
(x +
8 )( x -
8 )
Now Try These:
Factor the following polynomials and
check your work.
a. 4x2 – 9
b. 16x2 – 36
Answer: (2x + 3)(2x - 3)
Answer: (4x + 6)(4x - 6)
c. x2 – 4:
d. 25x2 – 16y2
Answer: (x - 2)(x + 2)
Answer: (5x – 4y)(5x + 4y)
Factoring by grouping
Examples of polynomials that are factored by
grouping are:
a. 6x2 + 3xy + 2xz + yz
b. 6x2 + 2xy – 3xz – yz
Note: You will see 4 terms when using the
grouping method.
Rules for Factoring by grouping:
-
-
-
Group terms so that
there is a common
factor in each
group.
Take out the
common factor in
both groups.
Combine like
groups.
Example:
10a2 + 2ab + 5ad + bd
I will group the first two terms and
the last two terms since both of
those groups contain a common
factor. Note: I am adding these
groups.
(10a2 + 2ab) + (5ad + bd)
Take out a common factor.
2a(5a + b) + d(5a + b)
Combine like groups:
( 2a + d )( 5a + b )
Examples of factoring by grouping:
Factor the polynomial:
a3 – 4a2 + 3a – 12
Factor the polynomial:
7ac2 + 2bc2 – 7ad2 – 2bd2
Group:
(a3 – 4a2)+ (3a – 12)
Factor:
a2 (a – 4) + 3(a – 4)
Combine:
(a2 + 3)(a – 4)
Group:
(7ac2 + 2bc2) + (– 7ad2 – 2bd2)
Factor:
c2(7a + 2b) + d2 (– 7a – 2b)
Factor a negative out of second
group:
c2(7a + 2b) - d2 ( 7a + 2b)
Now groups match – so,
Combine:
(c2 - d2) ( 7a + 2b)
Now Try These:
Factor the following polynomials and
check your work.
a. 6x2 + 3xy + 2xz + yz
Answer: (3x + z)(2x + y)
b. 6x2 + 2xy – 3xz – yz
Answer: (2x - z)(3x + y)
Factoring Completely
Some polynomials can be factored more
than once. This may not be apparent
from the beginning.
Just as integers can be factored into
primes, polynomials can too, and it may
take more than one step.
Rules for Factoring Completely
-
-
-
Factor a polynomial
using the
appropriate method.
Check each factor
to see if you can
factor it again.
If so, do it until all
polynomials are
prime.
Example:
3x2 – 21x + 30
Here, I notice that I have a
common factor of 3, so take
it out.
3(x2 – 7x + 10)
Now x2 – 7x + 10 can be
factored.
3(x – 5)(x – 2)
Now all terms are prime.
Now Try These:
Factor the following polynomials and
check your work.
a. 2x2 + 12x + 18
b. 3x2 – 21x – 54
Answer: 2(x + 3)(x + 3)
Answer: 3(x + 2)(x - 9)
c. 5x2 – 20:
d. 25x2 – 100y2
Answer: 5(x - 2)(x + 2)
Answer: 25(x – 2y)(x + 2y)
Part III
Warm Up Section 5-4 #3
Factor.
1. 7c3 – 28c2d + 35cd3
2. x2 – 5x – 14
3. x2 – 15x + 54
4. 3x2 – 22x + 35
5. 64x2 – 81
6. 3r + 3s + 5r3s + 5r2s2
Factor
The GCF
is 5ab.
Answer:
Distributive
Property
Factor
Answer:
Factor
Group to find the GCF.
Factor the GCF of
each binomial.
Answer:
Distributive Property
Factor
Answer:
Factor
To find the coefficient of the y terms, you must find two
numbers whose product is 3(–5) or –15 and whose sum
is –2. The two coefficients must be 3 and –5 since
and
.
Rewrite the expression using –5y and 3y in place
of –2y and factor by grouping.
Substitute –5y + 3y for –2y.
Associative Property
Factor out the GCF of each group.
Answer:
Distributive Property
Factor
Factor out the GCF.
Answer:
p2 – 9 is the difference
of two squares.
Factor
This is the sum of two cubes.
Sum of two cubes formula
with
and
Answer:
Simplify.
Factor
This polynomial could be considered the difference of
two squares or the difference of two cubes. The
difference of two squares should always be done before
the difference of two cubes.
Difference of two squares
Answer:
Sum and difference
of two cubes
Factor each polynomial.
a.
Answer:
b.
Answer:
c.
Answer:
d.
Answer:
Simplify
Factor the numerator
and the denominator.
Divide. Assume
a  –5, –2.
Answer: Therefore,
Simplify
Answer:
Example 1 Divide a Polynomial by a Monomial
Example 2 Division Algorithm
Example 3 Quotient with Remainder
Example 4 Synthetic Division
Example 5 Divisor with First Coefficient Other than 1
Steps for Dividing a Polynomial
by a Monomial
• 1. Divide each term of the polynomial by
the monomial.
a) Divide numbers
b) Subtract exponents
• 2. Remember to write the appropriate sign
in between the terms.
Example:
Answer:
Sum of quotients
Divide.
Answer:
Answer:
Try These
Divide the following polynomials.
1.
2.
Try These
Divide the following polynomials.
1.
x  2 xy  3x y
2
3
3
2.
2ab  3a b  5a
2 3
3
Use factoring to find
Answer:
Use factoring to find
Answer: x + 2
Try These
Divide the following polynomials by factoring.
1.
2.
3.
4.
Try These
Divide the following polynomials by factoring.
1.
2.
x2
3.
x2
4.
x3
x 5
Warm Up Section 5-5
Covering lessons 5.1-5.4
1. Simplify: 5x2y(4x4y3)
2. Simplify: (4a5b4c2) 3
6 3
18
x
y
3. Simplify:
4 8
12 x y
4. Multiply: (3x + 7)(x – 4)
5. Factor: x2 – 11x + 18
6. Factor: 3x2 + 7x – 20
Warm Up Section 5-5
Covering lessons 5.1-5.4
1. Simplify: 5x2y(4x4y3)
2. Simplify: (4a5b4c2) 3
6 3
18
x
y
3. Simplify:
4 8
12 x y
20x6y4
64a15b12c6
3x 2
5
2y
4. Multiply: (3x + 7)(x – 4) 3x2 – 5x – 28
5. Factor: x2 – 11x + 18
(x – 9)(x – 2)
6. Factor: 3x2 + 7x – 20
(3x – 5)(x + 4)
Example 1 Find Roots
Example 2 Simplify Using Absolute Value
Example 3 Approximate a Square Root
Simplifying Radicals
• When working with the simplification of
radicals you must remember some basic
information about perfect square
numbers.
Perfect Squares
1 =1x1
4 =2x2
9 =3x3
16 = 4 x 4
25 = 5 x 5
36 = 6 x 6
49 = 7 x 7
64 = 8 x 8
81 = 9 x 9
100 = 10 x 10
Perfect Squares Containing
Variables
a2 = a x a
a4 = a2 x a 2
a6 = a3 x a 3
a8 = a4 x a 4
a10 = a5 x a5
So, a variable is a
“perfect square” if it
has an even
exponent.
To take the square
root, just divide the
exponent by 2.
Simplifying Radical Expressions
To simplify means to find another
expression with the same value. It
does not mean to find a decimal
approximation.
Example:
and, although it is
equivalent to 5.65, we do not use the
decimal value since the radical value is
exact and the decimal is an estimate.
To simplify (or reduce) a radical:
• 1. Find the largest
perfect square which will
divide evenly into the
number under your
radical sign. This means
that when you divide, you
get no remainders, no
decimals, no fractions.
• 2. Write the number
appearing under your
radical as the product
(multiplication) of the
perfect square and your
answer from dividing.
• 3. Give each number in
the product its own
radical sign.
• 4. Reduce the "perfect"
radical which you have
now created.
Example:
• Reduce
:
the largest perfect square that divides evenly into 48 is
16.
• Find the largest perfect square which
will divide evenly into 48.
• Give each number in the product its own
radical sign.
Example Continued
• Reduce the "perfect" radical which you have
now created.
Simplify
Answer: The square roots of 16x6 are  4x3.
Simplify
Simplify
Answer: The fifth root is 3a2b3.
Simplify
Answer: You cannot take the square root of a negative num
Thus,
is not a real number.
Simplify.
a.
Answer:  3x4
b.
Answer:
c.
Answer: 2xy2
d.
Answer: not a real number
Try These
225
1
16
(5 g )
 (7)
2
8
169 x y
 27
4
0.25
4
3
4
z
8
(4 x  y )
2
Try These
225
 (7)
15
Not real #
1
16
1/4
(5 g )
25g2
2
 27
-3
4
0.25
z
8
z2
0.5
4
3
8
169 x y
13x4y2
4
(4 x  y )
4x - y
2
Simplify
Note that t is a sixth root of t6. The index is even, so the
principal root is nonnegative. Since t could be negative,
you must take the absolute value of t to identify the
principal root.
Answer:
Simplify
Since the index is odd, you do not need absolute value.
Answer:
Simplify.
a.
Answer:
b.
Answer:
Try These
 169
25
169
(2 x)
8
2
3
0.81
4
(4)
6
36 x y
10
125
z
12
(3x  6)
2
Try These
 169
-13
16x4
3
125
4
25
169
5/13
(2 x)
(4)
2
5
0.81
8
0.9
6
36 x y
6x3y5
4
z
12
z3
10
(3x  6)
3x+6
2
Assignment:
• P248 #40, 42, 46, 50
Example 1 Square Root of a Product
Example 2 Simplify Quotients
Example 3 Multiply Radicals
Example 4 Add and Subtract Radicals
Example 5 Multiply Radicals
Example 6 Use a Conjugate to Rationalize
a Denominator
Simplify
Factor into squares
where possible.
Product Property
of Radicals
Answer:
Simplify.
Simplify
Answer:
Simplify
Quotient Property
Factor into squares.
Product Property
Rationalize the denominator.
Answer:
Simplify
Quotient Property
Rationalize the denominator.
Product Property
Multiply.
Answer:
Simplify each expression.
a.
Answer:
b.
Answer:
Try These
72
4
32 x y
5
1 6 7
wz
32
3
5
3
4
54
16 y
4r 8
9
t
3
3
96
4 5
2 24m n
20 x 5
10
y
Try These
72
3
54
32 x y
5
3 2
4x2 y 2 2 y
5
1 6 7
wz
32 1 5
2
wz wz 2
3
16 y
3
96
24 6
3
6 2
4
4
3
4 5
2 24m n
2 y3 2
4r 8
4
9
2
r
t
t
t
5
4mn3 3mn2
20 x 5
10 2 x 2 5 x
y
y5
Warm Up 5-6
Simplify.
1. 36
2. 81x 2 y 6
3. 81x 6 y 7
4. 40 x 4 y11
5.
6.
3 5
3
24a b
4
32s 4t 7
Warm Up 5-6
Simplify.
1. 36
6
6
9xy
6
7
3
4
11
2. 81x y
3. 81x y
4. 40 x y
5.
6.
3
9x y
2
3
y
5
10 y
3
2
2x y
3 5
2ab 3b
4 7
4
24a b
4
3
2
32s t
2st 2t
3
Simplify
Product Property
of Radicals
Factor into cubes.
Product Property
of Radicals
Answer:
Multiply.
Simplify
Answer: 24a
Try These
1.
(3 12 )(2 21)
2.
(3 24 )(5 20)
Try These
1.
(3 12 )(2 21)
36 7
2.
(3 24 )(5 20)
 60 30
Simplify
Factor using squares.
Product Property
Multiply.
Combine like radicals.
Answer:
Simplify
Answer:
Try These
1.
12  48  27
2.
5 20  24  180  7 54
Try These
1.
12  48  27
3 3
2.
5 20  24  180  7 54
4 5  23 6
Simplify
F
O
I
L
Product Property
Answer:
Simplify
FOIL
Multiply.
Answer:
Subtract.
Simplify each expression.
a.
Answer:
b.
Answer: 41
Simplify
Multiply by
since
is the conjugate
of
FOIL
Difference of
squares
Multiply.
Answer:
Combine like terms.
Simplify
Answer:
Assignment
• P 254 #16-46 even
Example 1 Radical Form
Example 2 Exponential Form
Example 3 Evaluate Expressions with
Rational Exponents
Example 4 Rational Exponent with Numerator Other
Than 1
Example 5 Simplify Expressions with
Rational Exponents
Example 6 Simplify Radical Expressions
Write
Answer:
in radical form.
Definition of
Write
Answer:
in radical form.
Definition of
Write each expression in radical form.
a.
Answer:
b.
Answer:
Write
Answer:
using rational exponents.
Definition of
Write
Answer:
using rational exponents.
Definition of
Write each radical using rational exponents.
a.
Answer:
b.
Answer:
Evaluate
Method 1
Answer:
Simplify.
Method 2
Power of a Power
Multiply exponents.
Answer:
Evaluate
.
Method 1
Factor.
Power of a Power
Expand the square.
Find the fifth root.
Answer: The root is 4.
Method 2
Power of a Power
Multiply exponents.
Answer: The root is 4.
Evaluate each expression.
a.
Answer:
b.
Answer: 8
Weight Lifting The formula
can be used to estimate the maximum total mass that
a weight lifter of mass B kilograms can lift in two lifts,
the snatch and the clean and jerk, combined.
According to the formula, what is the maximum that
U.S. Weightlifter Oscar Chaplin III can lift if he weighs
77 kilograms?
Original formula
Use a calculator.
Answer: The formula predicts that he can
lift at most 372 kg.
Weight Lifting The formula
can be used to estimate the maximum total mass that
a weight lifter of mass B kilograms can lift in two lifts,
the snatch and the clean and jerk, combined.
Oscar Chaplin’s total in the 2000 Olympics was
355 kg. Compare this to the value predicted by the
formula.
Answer: The formula prediction is somewhat
higher than his actual total.
Weight Lifting Use the formula
where M is the maximum total mass that a weight
lifter of mass B kilograms can lift.
a. According to the formula, what is the maximum that a
weight lifter can lift if he weighs 80 kilograms?
Answer: 380 kg
b. If he actually lifted 379 kg, compare this to the value
predicted by the formula.
Answer: The formula prediction is slightly higher than his
actual total.
Simplify
.
Multiply powers.
Answer:
Add exponents.
Simplify
.
Multiply by
Answer:
Simplify each expression.
a.
Answer:
b.
Answer:
Simplify
.
Rational exponents
Power of a Power
Quotient of Powers
Answer:
Simplify.
Simplify
.
Rational exponents
Power of a Power
Multiply.
Answer:
Simplify.
Simplify
.
is the conjugate
of
Answer:
Multiply.
Simplify each expression.
a.
Answer: 1
b.
Answer:
c.
Answer:
Example 1 Solve a Radical Equation
Example 2 Extraneous Solution
Example 3 Cube Root Equation
Example 4 Radical Inequality
Solving Radical Equations
1.
2.
3.
Isolate the radical
Raise each side to the
appropriate power to
eliminate the radical.
Solve for the variable.
• Example:
Solve
for x.
1.
Isolate radical by adding 2 to both sides.
2.
Square both sides.
3.
So, x = 95
Solve
Original equation
Add 1 to each side to
isolate the radical.
Square each side to
eliminate the radical.
Find the squares.
Add 2 to each side.
Check
Original equation
Replace y with 38.
Simplify.
Answer: The solution checks. The solution is 38.
Solve
Answer: 67
Try These
Solve each equation.
Try These
Solve each equation.
25
144
1
-11
Solve
Original equation
Square each side.
Find the squares.
Isolate the radical.
Divide each side by –4.
Square each side.
Evaluate the squares.
Check
Original equation
Replace x with 16.
Simplify.
Evaluate the square roots.
Answer: The solution does not check, so there is
no real solution.
Solve
Answer: no real solution
.
Solve
In order to remove the
power, or cube root, you must
first isolate it and then raise each side of the equation to
the third power.
Original equation
Subtract 5 from each side.
Cube each side.
Evaluate the cubes.
Subtract 1 from each side.
Divide each side by 3.
Check
Original equation
Replace y with –42.
Simplify.
The cube root of –125 is –5.
Add.
Answer: The solution is –42.
Solve
Answer: 13
Try These
Solve each equation.
Try These
Solve each equation.
49
5
9
-20
Assignment
P 266 #16, 19, 22, 24
Example 1 Square Roots of Negative Numbers
Example 2 Multiply Pure Imaginary Numbers
Example 3 Simplify a Power of i
Example 4 Equation with Imaginary Solutions
Example 5 Equate Complex Numbers
Example 6 Add and Subtract Complex Numbers
Example 7 Multiply Complex Numbers
Example 8 Divide Complex Numbers
Keep in Mind:
• The square root of a negative number
does not exist.
• Example:
is not 5 or -5 since
5 x 5 = 25 and -5 x -5 = 25.
• So up until now, we could not simplify
.
i
i is defined to have the property that:
i2 = -1
therefore, we could say that square root
of -1 is i.
This allows us to simplify the square roots
of negative numbers such as
.
Examples
1. Simplify:
2. Simplify
6ix2
Since
is 5 and
is i, our answer is 5i.
Simplify
Answer:
.
Simplify
Answer:
.
Simplify.
a.
Answer:
b.
Answer:
Simplify
Answer:
.
=6
Simplify
Answer:
.
Simplify.
a.
Answer: –15
b.
Answer:
Simplify
Multiplying powers
Power of a Power
Answer:
Simplify
Answer: i
.
Solve
Original equation
Subtract 20 from
each side.
Divide each side by 5.
Take the square
root of each side.
Answer:
Solve
Answer:
Find the values of x and y that make the equation
true.
Set the real parts equal to each other and the imaginary
parts equal to each other.
Real parts
Divide each side by 2.
Imaginary parts
Answer:
Find the values of x and y that make the equation
true.
Answer:
Simplify
.
Commutative
and Associative
Properties
Answer:
Simplify
.
Commutative
and Associative
Properties
Answer:
Simplify.
a.
Answer:
b.
Answer:
Electricity In an AC circuit, the voltage E, current I,
and impedance Z are related by the formula
Find the voltage in a circuit with current 1 + 4 j amps
and impedance 3 – 6 j ohms.
Electricity formula
FOIL
Multiply.
Add.
Answer: The voltage is
volts.
Electricity In an AC circuit, the voltage E, current I,
and impedance Z are related by the formula E = I • Z.
Find the voltage in a circuit with current 1 – 3 j amps
and impedance 3 + 2 j ohms.
Answer: 9 – 7 j
Simplify
.
and
are conjugates.
Multiply.
Answer:
Standard form
Simplify
.
Multiply by
Multiply.
Answer:
Standard form
Simplify.
a.
Answer:
b.
Answer:
Explore online information about the
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Click on the Connect button to launch your browser
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Student Edition of your textbook. When you finish
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