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Chapter 4
4.1
-
Estimating Roots
4.2
-
Irrational Numbers
4.3
-
Mixed and Entire Radicals
4.4
-
Fractional Exponents and Radicals
4.5
-
Negative Exponents and Reciprocals
4.6
-
Applying the Exponent Laws
• Develop algebraic reasoning and number
sense.
• Demonstrate an understanding of irrational
numbers by representing, identifying,
simplifying, and ordering irrational numbers.
• Demonstrate an understanding of powers
with integral and rational exponents.
PERFORMANCE INDICATORS
AN02.01 Sort a set of numbers into rational and irrational numbers.
AN02.02 Determine an approximate value of a given irrational number.
AN02.03 Approximate the locations of irrational numbers on a number line,
using a variety of strategies, and explain the reasoning.
AN02.04 Order a set of irrational numbers on a number line.
AN02.05 Express a radical as a mixed radical in simplest form (limited to
numerical radicands).
AN02.06 Express a mixed radical as an entire radical (limited to numerical
radicands).
AN02.07 Explain, using examples, the meaning of the index of a radical.
AN02.08 Represent, using a graphic organizer, the relationship among the
subsets of the real numbers
(natural, whole, integer, rational, irrational).
KEY TERMS
• Real Number
• Natural Number
• Integer
• Rational Number
• Irrational Number
Real, R
Real numbers are all the numbers on the
continuous number line with no gaps.
Every decimal expansion is a real number.
Real numbers may be rational or irrational,
and algebraic or non-algebraic
(transcendental).
(π = 3.14159... and e = 2.71828... are
transcendental. )
A transcendental number can be defined
by an infinite series.
Natural Number, N
Natural numbers are the counting
numbers {1, 2, 3, ...} (positive integers)
or the whole numbers {0, 1, 2, 3, ...}
(the non-negative integers).
Mathematicians use the term "natural"
in both cases.
Integer, Z
Integers are the natural numbers and
their negatives
{... −3, −2, −1, 0, 1, 2, 3, ...}.
(Z is from German Zahl, "number".)
Rational, Q
Rational numbers are the ratios of
integers, also called fractions, such as
1/2 = 0.5 or 1/3 = 0.333...
Rational decimal expansions end or
repeat. (Q is from quotient.)
Radicals that are square
roots of perfect squares,
cube roots of perfect
cubes,
and so on are rational
numbers.
Rational numbers have
decimal
representations that
either terminate or
repeat.
Real Algebraic, AR
* The real subset of the algebraic
numbers: the real roots of
polynomials.
* Real algebraic numbers may be
rational or irrational. √2 = 1.41421... is
irrational.
* Irrational decimal expansions neither
end nor repeat.
• Answer the hand-out.
• Answer Practice Exercise.
PRACTICE EXERCISE
• Write your Chapter 4 vocabulary.
• Write sample problems in your
journal.
• Don’t forget to complete your Chapter
3 Media Project Reflection.
VOCABULARY
• entire
• mixed
• simplified
• approximate
• evaluate
• appropriate
QUICK REVIEW
WHAT ARE RADICALS?
•
A Radical is the root of a number.
radical sign
3 25
coefficient
radicand
•
For example:
WHAT ARE RADICALS?
•
A Radical is the root of a number.
•
For example:
25 = 5
3 25
coefficient
radicand
WHAT ARE RADICALS?
•
A Radical is the root of a number.
•
For example:
25 = 5
3 25
coefficient
radicand
5 is the square root of 25 because 5x5
= 25
WHAT ARE RADICALS?
•
A Radical is the root of a number.
•
For example:
25 = 5
3 25
5 is the square root of 25 because 5x5
=25
3
coefficient radicand
8=2
WHAT ARE RADICALS?
•
A Radical is the root of a number.
•
For example:
25  5
3 25
5 is the square root of 25 because 5x5
= 25
3
coefficient radical
82
2 is the cubed root of 8 because 2x2x2
= 8.
16  4
Please turn into pages 215-217 for examples of your textbook.
Please turn into pages 213-214 of your textbook and perform the
activity.
Please answer # 4 -14 of your textbook, pages 217-218.
1) Please copy Checkpoint 1 diagram on page 220.
2) Please answer # 15-25 of your textbook, pages 218-219.
3) Homework: Please answer “Assess Your Understanding”
Questions #1-11 on page 221.
CHAPTER 4 MEDIA PROJECT
WORK PERIOD: December 13, 2016
Deadline: Friday December 19-20, 2016
• Unit 2 – Video Project
• Part 2 - Chapter 4
Task :
• To record and create a video about Law of Exponents and Radicals
Curriculum Outcomes:
• * Express powers with rational exponents as radicals and vice versa,
when m and n are natural numbers, and x is a rational number.
𝑚
𝑛
1
𝑛
𝑚
𝑚
𝑛
1
𝑚𝑛
𝑛
• 𝑥 = (𝑥 )𝑚 = 𝑛 𝑥 and 𝑥 = 𝑥 = 𝑥 𝑚
• * Apply the following exponent laws to expressions with rational and
variable bases and integral and rational exponents, and explain the
reasoning.
Students’ Instruction
•
1) This task is by pair. Pair assignments will be based on performance standing in Q1.
•
2) Each pair will select a topic to explain. You must ask for the teacher’s approval.
•
3) You must create two outputs –
1
𝑛
•
Part 1 - on how to evaluate powers (Law of Exponents) of the form 𝑎
•
Part 2 - on how to simplify algebraic expressions with rational exponents
•
4) The criteria are:
Math Content
-
15 marks
Delivery
-
10 marks
•
Math Content – Explains how to solve in details and easy to understand. At least one key
term must be present in the lecture video.
•
Delivery- This involves the pronunciation of the presenter, spelling, and the English
grammar involved. It also includes how the output is presented (storyline, creativity,
originality) and the techniques and quality of the video.
•
Note: Answers should be correct. No mark for wrong answers.
PERFORMANCE INDICATORS
AN03.01 Explain, using patterns, why 𝒂−𝒏 =
𝟏
𝟏
𝒂𝒏
, 𝒂 ≠ 𝟎.
AN03.02 Explain, using patterns, why 𝒂𝒏 = 𝒏 𝒂 , 𝒏 > 𝟎.
AN03.03 Apply the following exponent laws to expressions with rational and variable bases and
integral and rational exponents, and explain the reasoning.
𝒂𝒎 𝒂𝒏 = 𝒂𝒎𝒏
𝒂𝒎 ÷ 𝒂𝒏 = 𝒂𝒎−𝒏 , 𝒂 ≠ 𝟎
(𝒂𝒎)𝒏 = 𝒂𝒎𝒏
(𝒂𝒃)𝒎 = 𝒂𝒎 𝒃𝒎
𝒂 𝒏 𝒂𝒏
( ) = 𝒏 ,𝒃 ≠ 𝟎
𝒃
𝒃
AN03.04 Express powers with rational exponents as radicals and vice versa, when m and n are
natural
numbers, and x is a rational number.
𝒎
𝒏
𝟏
𝒏
𝒎
𝒙 = (𝒙 ) =
𝒏
𝒎
𝒎
𝒏
𝒙 and𝒙 = 𝒙
𝟏
𝒎𝒏
=
𝒏
𝒙𝒎
AN03.05 Solve a problem that involves exponent laws or radicals.
AN03.06 Identify and correct errors in a simplification of an expression that involves powers.
EXAMPLE 4
REMEMBER!
CLASSWORK
• Answer # 4-16 on page 227.
HOMEWORK
• Answer # 17-22 on page 228.
REFERENCES
http://www.regentsprep.org/regents/math/algebra/aop1/lrat.htm
http://cnx.org/content/m38348/latest/
http://thinkzone.wlonk.com/Numbers/NumberSets.htm
http://www.mathsisfun.com/algebra/exponent-laws.html
http://www.math-for-all-grades.com/exponents-1.html
EXTRA NOTES
EXTRA LESSON
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
8
=
4*2 =
2 2
20
=
4*5
=
2 5
32
=
16 * 2 =
4 2
75
=
25 * 3 =
5 3
40
=
4 *10 = 2 10
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
48
=
16 * 3 =
4 3
80
=
16 * 5 =
4 5
50
=
25 * 2 =
5 2
125
=
25 * 5 =
5 5
450
=
225 * 2 = 15 2
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
18
=
=
288
=
=
75
=
=
24
=
=
72
=
=
ADDING AND SUBTRACTING RADICALS
• Only like radicals can be
added or subtracted.
• Like radicals can have
different coefficients as
long as the radicals are the
same.
• Subtract the coefficients
but keep the same radical.
• You may have to simplify
your question first, to get a
like radical.
ADDING AND SUBTRACTING RADICALS
• Only like radicals can be Example 1:
added or subtracted.
• Like radicals can have
different coefficients as
long as the radicals are the
same.
• Subtract the coefficients
but keep the same radical.
• You may have to simplify
your question first, to get a
like radical.
ADDING AND SUBTRACTING RADICALS
• Only like radicals can be Example 1:
added or subtracted.
• Like radicals can have
different coefficients as
long as the radicals are the
same.
• Subtract the coefficients
but keep the same radical.
• You may have to simplify
your question first, to get a
like radical.
4 3-2 3 = 2 3
ADDING AND SUBTRACTING RADICALS
• Only like radicals can be Example 1:
added or subtracted.
• Like radicals can have
different coefficients as
long as the radicals are the Example 2:
same.
• Subtract the coefficients
but keep the same radical.
• You may have to simplify
your question first, to get a
like radical.
4 3-2 3 = 2 3
ADDING AND SUBTRACTING RADICALS
• Only like radicals can be Example 1:
added or subtracted.
• Like radicals can have
different coefficients as
long as the radicals are the Example 2:
same.
• Subtract the coefficients
but keep the same radical.
• You may have to simplify
your question first, to get a
like radical.
4 3-2 3 = 2 3
4 3+2 3 =6 3
ADDING AND SUBTRACTING RADICALS
• Only like radicals can be Example 1:
added or subtracted.
• Like radicals can have
different coefficients as
long as the radicals are the Example 2:
same.
• Subtract the coefficients
but keep the same radical. Example 3:
• You may have to simplify
your question first, to get a
like radical.
4 3-2 3 = 2 3
4 3+2 3 =6 3
ADDING AND SUBTRACTING RADICALS
• Only like radicals can be Example 1:
added or subtracted.
• Like radicals can have
different coefficients as
long as the radicals are the Example 2:
same.
• Subtract the coefficients
but keep the same radical. Example 3:
• You may have to simplify
2 2 +2
your question first, to get a
2 2 +2
like radical.
4 3-2 3 = 2 3
4 3+2 3 =6 3
2
2 +4
6
2
8
4
2
2
COMMON MISTAKES MADE WHILE
ADDING AND SUBTRACTING RADICALS
•
One common mistake is adding
without having like radicals.
COMMON MISTAKES MADE WHILE
ADDING AND SUBTRACTING RADICALS
• One common mistake is adding without having
like radicals.
3 3 + 2 7 ¹ 5 10
COMMON MISTAKES MADE WHILE
ADDING AND SUBTRACTING RADICALS
• One common mistake is adding without having
like radicals.
• Always make sure that the radicals are similar.
3 3 + 2 7 ¹ 5 10
COMMON MISTAKES MADE WHILE
ADDING AND SUBTRACTING RADICALS
•
One common mistake is adding
without having like radicals.
•
Always make sure that the radicals
are similar.
•
Another common mistake is
forgetting to simplify.
3 3 + 2 7 ¹ 5 10
COMMON MISTAKES MADE WHILE
ADDING AND SUBTRACTING RADICALS
•
One common mistake is adding
without having like radicals.
•
Always make sure that the radicals
are similar.
3 3 + 2 7 ¹ 5 10
•
Another common mistake is
forgetting to simplify.
8+2 8 =3 8
COMMON MISTAKES MADE WHILE
ADDING AND SUBTRACTING RADICALS
•
One common mistake is adding
without having like radicals.
•
Another common mistake is
forgetting to simplify.
•
Always make sure that the radicals
are similar.
•
This question is not done! The
answer is:
3 3 + 2 7 ¹ 5 10
6 2
8+2 8 =3 8
+
To combine radicals: combine the
coefficients of like radicals
SIMPLIFY EACH EXPRESSION:
6 7 5 7 3 7 
5 6 3 7 4 7 2 6 
8 7
3 6 7 7
COMBINED OPERATIONS WITH
RADICALS
•
You follow the same steps with
these as you do with polynomials.
COMBINED OPERATIONS WITH
RADICALS
•
You follow the same steps with
these as you do with polynomials.
•
Use the distribution property.
COMBINED OPERATIONS WITH
RADICALS
•
You follow the same steps with
these as you do with polynomials.
•
Use the distribution property.
•
Example:
COMBINED OPERATIONS WITH
RADICALS
•
You follow the same steps with
these as you do with polynomials.
•
Use the distribution property.
•
Example:
2 ( 10 - 2 )
COMBINED OPERATIONS WITH
RADICALS
•
You follow the same steps with
these as you do with polynomials.
•
Use the distribution property.
•
Example:
2 ( 10 - 2 )
20 - 4
COMBINED OPERATIONS WITH
RADICALS
•
You follow the same steps with
these as you do with polynomials.
•
Use the distribution property.
•
Example:
2 ( 10 - 2 )
20 - 4
2 5-2
COMMON MISTAKES MADE WHILE
DOING COMBINED OPERATIONS
•
Be careful to multiply correctly.
COMMON MISTAKES MADE WHILE
DOING COMBINED OPERATIONS
•
Be careful to multiply correctly.
•
Correct Answer:
COMMON MISTAKES MADE WHILE
DOING COMBINED OPERATIONS
•
Be careful to multiply correctly.
•
Correct Answer:
(3 - 3 )( 4 + 3 )
12 + 3 3 - 4 3 - 3
9- 3
COMMON MISTAKES MADE WHILE
DOING COMBINED OPERATIONS
•
Be careful to multiply correctly.
•
Correct Answer:
(3 - 3 )( 4 + 3 )
12 + 3 3 - 4 3 - 3
9- 3
•
Incorrect Answer:
COMMON MISTAKES MADE WHILE
DOING COMBINED OPERATIONS
•
Be careful to multiply correctly.
•
Correct Answer:
•
Incorrect Answer:
(3 - 3 )( 4 + 3 )
(3 - 3 )( 4 + 3 )
12 + 9 - 4 3 - 3
12 + 3 3 - 4 3 - 3 12 - 4 3
9- 3
SIMPLIFY EACH EXPRESSION:
SIMPLIFY EACH RADICAL FIRST
AND THEN COMBINE.
2 50  3 32  2 25 * 2  3 16 * 2 
2 * 5 2  3* 4 2 
10 2  12 2 
2 2
SIMPLIFY EACH EXPRESSION:
SIMPLIFY EACH RADICAL FIRST AND
THEN COMBINE.
3 27  5 48  3 9 * 3  5 16 * 3 
3*3 3  5* 4 3 
9 3  20 3 
29 3
LETS DO SOME WORK!
Simplify each expression.
6 5  5 20 
18  7 32 
2 28  7  6 63 
6 5 5 6 3 6 
3 24  7 54 
2 8  7 32 
MULTIPLYING RADICALS
Steps
Example:
MULTIPLYING RADICALS
Steps
Example:
3 3´4 3
MULTIPLYING RADICALS
Steps
1.
Multiply coefficients together
Example:
3 3´4 3
1. 12
3´ 3
MULTIPLYING RADICALS
Steps
1.
Multiply coefficients together
2.
Multiply radicals together
Example:
3 3´4 3
1. 12
2.
3´ 3
12 ´ 9
MULTIPLYING RADICALS
Steps
1.
Multiply coefficients together
2.
Multiply radicals together
3.
Simplify radicals if possible
Example:
3 3´4 3
12 3 ´ 3
12 ´ 9
12 ´ 3 = 36
EXAMPLES OF MULTIPLYING
RADICALS
•
Correct example:
EXAMPLES OF MULTIPLYING
RADICALS
•
Correct example:
5 3 ´3 5
EXAMPLES OF MULTIPLYING
RADICALS
•
Correct example:
5 3 ´3 5
15 15
EXAMPLES OF MULTIPLYING
RADICALS
•
Correct example:
5 3 ´3 5
•
Be careful to multiply the
coefficients with coefficients and
radicals with radicals.
15 15
EXAMPLES OF MULTIPLYING
RADICALS
•
Correct example:
5 3 ´3 5
•
Be careful to multiply the
coefficients with coefficients and
radicals with radicals.
15 15
•
Incorrect Example:
EXAMPLES OF MULTIPLYING
RADICALS
•
Correct example:
5 3 ´3 5
•
Be careful to multiply the
coefficients with coefficients and
radicals with radicals.
15 15
•
Incorrect Example:
3 7 ´5 2
21 ´ 10
210
EXAMPLES OF MULTIPLYING
RADICALS
•
Correct example:
•
Incorrect Example:
5 3 ´3 5
•
3
7
´
5
2
Don’t multiply
Be careful to multiply the
these
coefficients with coefficients and
radicals with radicals.
together!
•
15 15
21 ´ 10
This is Wrong!
210
´
DIVIDING RADICALS (PART 1)
•
There are 4 steps to dividing
radicals.
DIVIDING RADICALS (PART 1)
•
There are 4 steps to dividing
radicals.
•
1. Reduce coefficients and/or
radicals by common factor (if you
can).
DIVIDING RADICALS (PART 1)
•
There are 4 steps to dividing
radicals.
•
1. Reduce coefficients and/or
radicals by common factor (if you
can).
•
2.Take out any perfect squares (if
possible).
DIVIDING RADICALS (PART 1)
• There are 4 steps to
dividing radicals.
• 1. Reduce coefficients
and/or radicals by
common factor (if you
can).
• 2.Take out any perfect
squares (if possible).
• Rationalize denominators
DIVIDING RADICALS (PART 1)
• There are 4 steps to
dividing radicals.
• 1. Reduce coefficients
and/or radicals by
common factor (if you
can).
• 2.Take out any perfect
squares (if possible).
• Rationalize denominators.
• Reduce coefficients
again.
DIVIDING RADICALS (PART 1)
• There are 4 steps to
dividing radicals.
• 1. Reduce coefficients
and/or radicals by common
factor (if you can).
• 2.Take out any perfect
squares (if possible).
• 3. Rationalize
denominators.
• 4. Reduce coefficients
again.
•
Example:
DIVIDING RADICALS (PART 1)
• There are 4 steps to
dividing radicals.
• 1. Reduce coefficients
and/or radicals by common
factor (if you can).
• 2.Take out any perfect
squares (if possible).
• 3. Rationalize
denominators.
• 4. Reduce coefficients
again.
•
Example:
2 15 2 5
=
=2 5
1
3
DIVIDING RADICALS (PART 1)
• There are 4 steps to
dividing radicals.
• 1. Reduce coefficients
and/or radicals by common
factor (if you can).
• 2.Take out any perfect
squares (if possible).
• 3. Rationalize
denominators.
• 4. Reduce coefficients
again.
•
Example:
2 15 2 5
=
=2 5
1
3
Divide top and
bottom by 3
COMMON ERRORS DONE WHILE
DIVIDING RADICALS
•
Be careful not to divide a
coefficient by a radical.
COMMON ERRORS DONE WHILE
DIVIDING RADICALS
•
Be careful not to divide a
coefficient by a radical.
•
Example of this error:
COMMON ERRORS DONE WHILE
DIVIDING RADICALS
•
Be careful not to divide a
coefficient by a radical.
•
Example of this error:
12
¹ 4
3
COMMON ERRORS DONE WHILE
DIVIDING RADICALS
•
Be careful not to divide a
coefficient by a radical.
•
Example of this error:
12
¹ 4
3
•
The correct answer to this question
is:
COMMON ERRORS DONE WHILE
DIVIDING RADICALS
•
Be careful not to divide a
coefficient by a radical.
•
Example of this error:
12
¹ 4
3
•
The correct answer to this question
is:
12 12 3
=
=4 3
3
3
COMMON ERRORS DONE WHILE
DIVIDING RADICALS
•
Be careful not to divide a
coefficient by a radical.
•
Example of this error:
12
¹ 4
3
•
The correct answer to this question
is:
12 12 3
=
=4 3
3
3
Multiply top and
bottom by
3
DIVIDING RADICALS (PART 2)
•
When rationalizing binomial
denominators, multiply both sides
by the conjugate.
DIVIDING RADICALS (PART 2)
•
When rationalizing binomial
denominators, multiply top and
bottom by the conjugate.
(3+ 3)
(3 - 3)
DIVIDING RADICALS (PART 2)
•
When rationalizing binomial
denominators, multiply top and
bottom by the conjugate.
(3+ 3)
(3 - 3)
Binomial denominator.
DIVIDING RADICALS (PART 2)
•
When rationalizing binomial
denominators, multiply top and
bottom by the conjugate.
(3+ 3)
(3- 3)
Binomial denominator.
•
Example:
DIVIDING RADICALS (PART 2)
•
When rationalizing binomial
denominators, multiply top and
bottom by the conjugate.
(3+ 3)
(3 - 3)
Binomial denominator.
•
Example:
( 3 + 3 ) (3 + 3 )
( 3 - 3 ) (3 + 3 )
DIVIDING RADICALS (PART 2)
•
When rationalizing binomial
denominators, multiply top and
bottom by the conjugate.
(3+ 3)
(3 - 3)
Binomial denominator.
•
Example:
( 3 + 3 ) (3 + 3 )
( 3 - 3 ) (3 + 3 )
Multiply numerator
and denominator by
(3 + 3 )
DIVIDING RADICALS (PART 2)
•
When rationalizing binomial
denominators, multiply top and
bottom by the conjugate.
(3+ 3)
(3 - 3)
Binomial denominator.
•
Example:
( 3 + 3 ) (3 + 3 )
( 3 - 3 ) (3 + 3 )
9+6 3+3
9-3
Multiply
top and
bottom by
(3 + 3 )
DIVIDING RADICALS (PART 2)
•
When rationalizing binomial
denominators, multiply top and
bottom by the conjugate.
(3+ 3)
(3 - 3)
Binomial denominator.
•
Example:
( 3 + 3 ) (3 + 3 )
( 3 - 3 ) (3 + 3 )
9+6 3+3
9-3
2+ 3
Multiply
top and
bottom
(3 + 3 )
EXTRA PRACTICE FOR QUIZ!
Solve without a
calculator.
1.
25 =
2.
1.21 =
Use your calculator
to solve.
3.
242064 =
4.
3.375 =
3
Simplify.
96 =
5.
3
6. 3
81 =
Extra Practice for Quiz!
Solve.
Multiply.
7. 2 8 ´ 3 2 =
8. 16 ´ 54 =
3
3
Add or Subtract.
9. 3 16 + 53 54 =
10. 3 32 - 4 2 =
11. (2 6 - 3 8 )
2
Divide.
12. 36 =
2
13. ( 2 - 3 ) =
( 2 + 3)
ANSWERS!
3
1. 5
6. 9 3
2. 1.1
7. 24
3. 492
8. 63 4
4. 1.5
9. 173 2
5. 4 6
10. 8 2
Answers!
11.
96 - 48 3
12. 3 2
5-2 6
13.
-1