Big Ideas, and New Vocabulary
Transcript of Big Ideas, and New Vocabulary
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Math 1201
Unit 4: Roots & Powers
Read Building On, Big Ideas, and New Vocabulary, p. 202 text.
Ch. 4 Notes
§4.1 Estimating Roots (1 class)
Read Lesson Focus p. 4 text.
Outcomes
1. Define and give an example of a radical. pp. 204, 539
2. Identify the index and the radicand of a radical. p. 204
3. Determine the exact value, or an approximation of the exact value, of the root of a number. p.
205
nDef : A radical is an expression consisting of a radical sign (radical symbol), a radicand, and an
index.
E.g.: 3 52 4 2 327
9 9; 25; 100; 110; 0.25 0.25;81
E.g.: Complete the table below.
General Form of Radical Example Index Radicand 2x x (Square Root) 29 9 2 9
2x x 20.25 0.25 0.25
2x x 2
36 36
81 81 2
3 x (Cube Root) 3 27 3 27
3 x 3 0.001 3
2
3 x 3
64
125
64
125
4 x (Fourth Root) 4 126 126
4 x 4 12.58 4
4 x 4
16
625
16
625
5 x (Fifth Root) 5 18
5 x 5 0.00056
5 x 5
22
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Evaluating Radicals
Evaluating a radical means to write it as an exact or as an approximate value.
Complete the following table.
Radical Value Exact? or
Approximate?
Why that Value?
Because … 2 9 or 9 3 Exact 23 9
2 18 or 18 4.242640687… Approximate 2
4.242640687 18
2 18 or 18 ???????? ????????
2 0.25 or 0.25 0.5 Exact 2
0.5 0.25
27 7
or 11 11
0.7977240352… Approximate 2 7
0.797724035211
21 1
or 9 9
1
or 0.33
Exact
21 1
3 9
3 16
3 27 Exact
3 0.64
3 0.64
316
81
4 16
4 16
4 27
416
81
4 0.64
5 32 Exact
3
5 32 5
2 32
53
8 Approximate
E.g.: Evaluate to two decimal places, if necessary and if possible. Explain why your answer is correct.
The first four are done for you.
a) 250
250 15.81 because 2
15.81 250
b) 121
121 11 because 2
11 121
c) 36
36 ??????????
d) 3 85
3 85 4.40 because 3
4.40 85
e) 3 1728
3 1728 _____________ because 3
___________ 1728
f) 3 915.0625
3 915.0625 _____________ because 3
___________ 915.0625
g) 3 0.000027
3 0.000027 _____________ because 3
___________ 0.000027
h) 4 92.3521
4 92.3521 _____________ because 4
___________ 92.3521
i) 4 625
4 625 ??????????
j) 5 243
4
5 243 _____________ because 5
___________ 243
k) 532
3125
532
_____________3125 because
5
32
_____________ 3125
Radicals with Negative Radicands
*******Negative radicands can only be evaluated if the INDEX is ODD.
Radicals with negative radicands cannot be evaluated if the index is EVEN. This is because no number
can be multiplied by itself an even number of times to give a negative answer.
E.g.: Determine if it is possible to evaluate each of the following radicals.
a) 300 Y OR N
b) 300 Y OR N
c) 3 300 Y OR N
d) 3 300 Y OR N
e) 4 300 Y OR N
f) 4 300 Y OR N
g) 5 300 Y OR N
h) 5 300 Y OR N
E.g.: Evaluate each of the following radicals rounded correctly to FOUR decimal places, if necessary
and if possible.
a) 3 300 _______________
b) 3 15.62590 _______________
c) 3 27000 _______________
d) 3 300 _______________
e) 38
_____________27
f) 3 15625 _______________
g) 3 16 _______________
h) 4 16 _______________
i) 4 16 _______________
j) 4 16 _______________
5
k) 532
_______________3125
l) 5 3125 _______________
m) 6 3125 _______________
E.g.: For each given number, write an equivalent form as a radical.
Number Equivalent Square
Root
Equivalent Cube
Root
Equivalent Fourth
Root
Equivalent Fifth Root
5 25 3 125 5 3125
0.8 0.64 3 0.512 4 0.4096
-7 Not possible Not possible 5 16807
2
7
3
8
343 4
16
2401 5
32
16807
-0.25 Not possible 3 0.015625 5 0.00009765625
E.g.: Choose values for x and n so that n x is:
a) A whole number 0,1,2,3,4,5,6,
Possible Answers: 3 5425 5; 27 3; 625 5; 32768 8
b) A negative integer 4, 3, 2, 1
Possible Answers: 3 527 3; 32768 8
c) A rational number (a fraction, a decimal that ends, or a decimal that repeats)
Possible Answers: 3 4121 11 1
; 76.765625 4.25; 0.164 8 6561
d) An irrational number (an approximate decimal)
Possible Answers:
5432
27 5.196152423 ; 0.8735804647 ; 1.331335364 ; 33 2.0123466173
Do #’s 1, 2, 4 c, 5 a, d, e, f, 6, p. 206 text in your homework booklet.
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§4.2 The Real Number System (1 class)
Read Lesson Focus p. 207 text.
Outcomes
1. Explain what is meant by classifying numbers. See notes
2. Classify real numbers. p. 209
3. Define and give examples of irrational numbers. pp. 208, 537
4. Give the symbol used to represent the irrational numbers. See notes
5. Order irrational numbers. pp. 209-210
nDef : Classifying numbers means determining to what set(s) of numbers a given number belongs. In
order to classify numbers, you need to know the different sets of numbers.
Sets of Numbers
1. Natural Numbers : The counting numbers. 1,2,3,4,5,6,7,
2. Whole Numbers : The counting numbers plus zero. 0,1,2,3,4,5,6,7,
3. Integers or : The whole numbers plus the opposites of the natural numbers.
or 7, 6, 5, 4, 3, 2, 1,0,1,2,3,4,5,6,7,
4. Rational Numbers : Numbers that can be written as fractions OR decimals that repeat OR
decimals that end. E.g.: 3
, 4.75, 1.37
. , , , 0
aa b b
b
5. Irrational Numbers or : Numbers that CANNOT be written as fractions OR decimals
that end OR decimals that repeat. E.g.: 3 4 52
2, 5.764, 71, , ,3
e .
, , , 0a
a b bb
6. Real Numbers : The rational numbers and the irrational numbers combined.
7. Complex Numbers : Numbers that can be written in the form a bi where ,a b and
1i . E.g.: 4 42, 15, 3 4 ,
9i i
These numbers can be represents in a Venn diagram (see below).
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Important observations you need to make from the chart.
Observation #1:
Notice that 9 is a natural number. It is because 9 3 .
Observation #2:
Notice that the only difference between natural numbers and whole numbers is the zero.
Whole numbers = Natural numbers + zero
Observation #3:
Notice that the difference between whole numbers and integers are the negative numbers.
Integers = Whole numbers + the negatives of the whole numbers
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Observation #4:
All integers are fractions. Not all fractions are integers.
E.g.: -2 is an integer and can be written as 2 2 2
or or 1 1 1
to make it a fraction.
However, 1
0.33333333333333 is not an integer.
Observation #5:
Fractions can be written as a terminating decimal or a repeating decimal
E.g.: 1
0.52 and 0.5 is a terminating decimal.
10.3333333333333 0.3
3 is a repeating decimal.
Observation #6:
Rational numbers = Integers + fractions
Observation #7:
Irrational numbers are numbers that cannot be written as a fraction.
E.g.: , 7
Another way to see them is that they are neither repeating decimals nor terminating decimals.
Observation #8:
Real numbers = rational numbers + irrational numbers. Every number you know of is a real number.
Observation #9:
The difference between complex numbers and real numbers is that complex numbers give solutions for
the following expressions and more! Every number is a complex number.
7, 1 9, 25 5i
Complete the following table. The first one and the last one are done for you.
Number W I or
4.5
121
2
5
9
0
0.313233…
3
-6
Ordering Irrational Numbers
Change each number to a decimal approximation.
E.g.: Order 3 3 42, 2, 6, 11, 40 from least to greatest.
3
3
4
2 1.4142
2 1.2599
6 1.8171
11 3.3166
40 2.5149
So the order from least to greatest is 3 3 42, 2, 6, 40, 11 .
Do #’s 3, 4, 7, 11, 14, 15, p. 211 text in your homework booklet.
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§4.3 Mixed and Entire Radicals (2 classes)
Read Lesson Focus p. 213 text.
Outcomes
1. Simplify radicals. pp. 215-216
2. Define and give an example of an entire radical. pp. 217, 536
3. Define and give an example of a mixed radical. pp. 217, 538
4. Rewrite an entire radical as a mixed radical. p. 214, 216
5. Rewrite a mixed radical as an entire radical. p. 214, 217
To help simplify radicals, we need to use a special property of radicals and some special numbers (i.e.
perfect squares, perfect cubes, and so on).
Multiplication Property of Radicals
Determine if the following expressions are equal.
Expression 1 Expression 2 Expression 3 Equal? Y or N
12 4 3 4 3 Y
99 9 11 9 11
32 16 2 16 2
3 24 3 8 3 3 38 3
3 54 3 27 2 33 27 2
3 256 3 64 4 33 64 4
4 32 4 16 2 44 16 2
4 243 4 81 3 4 481 3
4 512 4 256 2 44 256 2
All the examples in the table above illustrate the Multiplication Property of Radicals.
***** n n na b a b
Note that this property works both ways:
i. n n na b a b 33 354 27 2 and
ii. n n na b a b 33 327 2 54
Simplifying Radicals
To simplify radicals, we will use the Multiplication Property of Radicals and some special numbers.
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Special Numbers used to Simplify Radicals
Complete the table below. Some have been done for you.
Perfect Squares
(Used to simplify Square
Roots)
Perfect Cubes
(Used to simplify Cube
Roots)
Perfect Fourths
(Used to simplify
Fourth Roots)
Perfect Fifths
(Used to simplify Fifth
Roots) 21 1 31 1 41 1 51 1 22 4 32 8 42 16 52 32 23 9 33 27 43 81
53 243
310 1000 410 10000 510 100 000
220 400
We are going to use the Multiplication Property of Radicals and the special numbers in the table to
simplify radicals. The special numbers that we use depend on whether we are simplifying a square root,
a cube root, a fourth root, or a fifth root.
E.g.: Simplify 99 .
Since we are dealing with a square root, we look in our table for the BIGGEST perfect square that
divides into 99. That perfect square is 9. Using the Multiplication Property of Radicals gives
99 9 11 3 11
E.g.: Simplify 3 54 .
Since we are dealing with a cube root, we look in our table for the BIGGEST perfect cube that divides
into 54. That perfect cube is 27. Using the Multiplication Property of Radicals gives
3 33 354 27 2 3 2
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E.g.: Simplify 4 48 .
Since we are dealing with a fourth root, we look in our table for the BIGGEST perfect fourth that
divides into 48. That perfect fourth is 16. Using the Multiplication Property of Radicals gives
4 4 4 448 16 3 2 3
E.g.: Simplify 5 96 .
Since we are dealing with a fifth root, we look in our table for the BIGGEST perfect fifth that divides
into 96. That perfect fifth is 32. Using the Multiplication Property of Radicals gives
5 5 5 596 32 3 2 3
How do you know when to stop? You stop when the no number in the column you are using divides
evenly into the radicand.
nDef : An entire radical is a radical of the form n x .
E.g.: 99 , 3 54 , 4 48 , 5 96
nDef : A mixed radical is a radical of the form na x , where 1a .
E.g.: 3 11 , 33 2 , 42 3 , 52 3
E.g.: Write 48 as a mixed radical.
48 16 3 16 3 4 3
E.g.: Write 4 3 as an entire radical. Since 16 4 , we can write
4 3 16 3 16 3 48
E.g.: Write 3 54 as a mixed radical.
3 33 3 354 27 2 27 2 3 2
E.g.: Write 33 2 as an entire radical. Since 3 27 3 , we can write
3 33 3 33 2 27 2 27 2 54
E.g.: Write 4 48 as a mixed radical.
4 4 4 4 448 16 3 16 3 2 3
E.g.: Write 42 3 as an entire radical. Since 4 16 2 , we can write
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4 4 4 4 42 3 16 3 16 3 48
E.g.: Write 5 96 as a mixed radical.
5 5 5 5 596 32 3 32 3 2 3
E.g.: Write 52 3 as an entire radical. Since 5 32 2 , we can write
5 5 5 5 52 3 32 3 32 3 96
Complete the table below. Some have been done for you.
Entire Radical n a b n na b Mixed Radical
75 25 3 25 3 5 3
45
72
3 16 3 8 2 33 8 2 32 2
3 81
3 250
4 160 4 16 10 4 416 10 42 10
4 80
4 512
Complete the table below. Some have been done for you.
Mixed Radical n na b n a b Entire Radical
6 3 36 3 36 3 108
4 5
7 10
32 5 3 38 5 3 8 5 3 40 33 4
35 7
43 4 44 81 4 4 81 4 4 324
43 8
45 5
Do #’s 4, 5, 9, 10 a,c,e,h, 11 a,c,e,g,i, 12 a,c,e,g,i, 14, 17 a,c, 18 a,c, 20, 21, pp. 218-219 text in your
homework booklet.
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§4.4 Fractional Exponents and Radicals (2 classes)
Read Lesson Focus p. 222 text.
Outcomes
1. Evaluate radicals with fractional exponents. pp. 223-226
2. Explain how 1
and n nx x are related. p. 223
3. Explain how and m
mn nx x are related. p. 225
4. Rewrite a radical as an equivalent power. p. 225
In this section you will convert radicals to an equivalent power, convert a power to an equivalent radical,
simplify powers which have exponents which are fractions, and extend some basic laws of exponents to
include exponents that are fractions.
The Relationship between the Radical n x and the Power
1
nx
Complete the following table.
Radical Power Radical Power Radical Power 2x x
1
2x Equal
Y or
N?
3 x 1
3x Equal
Y or
N?
4 x 1
4x Equal
Y or
N? 29 9
1
29 Y 3 64
1
364 Y 4 625
1
4625 Y
211 11 1
211 Y 3 12
1
312 Y 4 34
1
434 Y
245.678 45.678 1
245.678 Y 3 10.68
1
310.68 Y 4 1001.28
1
41001.28 Y
23 3
4 4
1
23
4
Y 3
11
4
1
311
4
Y 4
89
24
1
489
24
Y
3 8 1
38 Y Y
From the table above you should see that the radical n x is the same as the power
1
nx .
When n is a natural number (1, 2, 3, 4, …) and x is a rational number (decimal that ends or repeats), then
1
*************** n nx x
E.g.:
11 1 13
542 43 58 8 2
25 25 5; ; 5.0625 5.0625 1.5; 7776 7776 6125 125 5
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E.g.: Complete the following table.
Want do you think the exponent 1
2 means? The exponent
1
2 means the square root.
Want do you think the exponent 1
3 means?
Want do you think the exponent 1
4 means?
Want do you think the exponent 1
5 means?
Want do you think the exponent 1
n means? The exponent
1
n means the n
th root.
Recall that fractions such as 1 1 1
, , and 2 4 5
can be written as the terminating decimals 0.5, 0.25, and 0.2
This may be useful in question # 4, p. 227 text.
E.g.: Complete the table below. The first one is done for you.
Radical n x Equivalent Power
1
nx
2217 217 1
2217 3 50
4 25
5 30
E.g.: Evaluate each without using a calculator:
11 1 1 43 2 3
811000 ; 0.25 ; 8 ; ;
16
1
331000 1000 10 1
20.25 0.25 0.5
1
338 8 2
1
44
81 81 3
16 16 2
Do #’s 3, 4 a, b, d, 5, 6, 13 a, c, e, 14 b, c, d, p. 227 text in your homework booklet.
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The Relationship between the Radical m
n x and the Power
m
nx
Complete the following table.
Radical m
n x Power m
nx Equal Y or N?
3
4 16 3
416 Y
2
5 27 2
527 Y
4
32
5
4
32
5
Y
3
17.625 3
217.625 Y
From the table above you should see that the radical m
n x is the same as the power
m
nx .
When m and n are a natural numbers (1, 2, 3, 4, …) and x is a rational number (decimal that ends or
repeats), then
***************m
mn nx x
223 5
3 5342 43
225 5
8 8 4E.g.: 25 25 125; ; 5.0625 5.0625 7.59375
125 125 25
7776 7776 36
E.g.: Complete the table below.
Radical m
n x Equivalent Power
m
nx
4 32 1
432
3
4 5 0.755
2
526
5
315
3.56
11
0.7512
17
E.g.: Evaluate each without using a calculator:
33 3 34 32 0.44 2 43 2
8181 ; 0.01 ; 27 ; ; 32 ; 4 ; 16
16
3
334481 81 3 27
3 3 332
2 21 1 1 1
0.01 0.001100 100 10 1000
4 4 43327 27 3 81
3 3 3281 81 9 729
16 16 4 64
2 20.4 25532 32 32 2 4
3 3
24 4 impossible
3 3 3 344 416 1 16 1 16 1 2 1 8 8
E.g.: The value (V) of a car is given by the formula 232000 0.85t
V where t represents the age of the
car in years. Find the value of the car after 5 years.
If 5t then,
5
2.5232000 0.85 32000(0.85) $21315.59V
Do #’s 8, 10 a, b, c, e, 11, a, b, c, 12, 16 a, 17, 18, 19, 20 b, pp. 227-228 text in your homework
booklet.
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§4.5 Negative Exponents and Reciprocals (2 classes)
Read Lesson Focus p. 229 text.
Outcomes
1. Evaluate powers with negative exponents. pp. 231-232
2. Explain how 1
and n
nx
x
are related. p. 231
3. Explain how 1
and n
nx
x are related. p. 231
4. Explain how
n
p
q
and
n
q
p
are related pp. 231-232
In this section you will evaluate powers with negative exponents.
The Relationship between the Power nx and the Power 1
nx
Complete the following table.
nx 1
nx Equal Y or N?
25 2
1
5
3
4
3
1
4
43.25 4
1
3.25
52 5
1
2
From the table above you should see that the power nx is the same as the power 1
nx.
When n is a natural number (1, 2, 3, 4, …) and x is a rational number (decimal that ends or repeats), then
1************** n
nx
x
E.g.: 2 3 4 5
2 3 4 5
1 1 1 1 1 1 1 16 ; 5 ; 4 ; 3
6 36 5 125 4 256 3 243
Do #’s 3 a, c, 4, a, b, 5, 6, 9, a, b, c, d, 10, a, c, d, 13 a, b, c, p. 233 text in your homework booklet.
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The Relationship between the Power nx and the Power
1nx
Complete the following table.
nx 1
nx Equal Y or N?
25 2
1
5 Y
3
4
3
1
4
Y
43.25 4
1
3.25 Y
52 5
1
2 Y
From the table above you should see that the power nx is the same as the power
1nx
.
When n is a natural number (1, 2, 3, 4, …) and x is a rational number (decimal that ends or repeats), then
1************** n
nx
x
E.g.: 2 3 4 5
2 3 4 5
1 1 1 16 36; 5 125; 4 256; 3 243
6 5 4 3
Do #’s 3 d, 8 f, p. 233 text in your homework booklet.
The Relationship between the Power
n
p
q
and the Power
n
q
p
Complete the following table.
Power
n
p
q
Power
n
q
p
Equal Y or N?
24
3
23
4
Y
31
6
36
1
Y
42
5
45
2
Y
57
3
53
7
Y
20
From the table above you should see that the power
n
p
q
is the same as the power
n
q
p
.
When n is a natural number (1, 2, 3, 4, …), p and q are integers with 0q then
**************
n n
p q
q p
E.g.:
2 2 3 3 4 4 5 51 5 2 7 343 3 5 625 4 7 16807
25; ; ;5 1 7 2 8 5 3 81 7 4 1024
Do #’s 3 b,7, 8 d, e, 12, 13 d, e, f, p. 233 text in your homework booklet.
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§4.6 Applying the Laws of Exponents (2 classes)
Read Lesson Focus p. 237 text.
Outcomes
1. Simplify expressions using the laws of exponents. pp. 238-241
In this section you will review the laws of exponents and use them to simplify expressions containing
variables.
Recall the following laws of exponents from previous math courses.
Exponent Law Exponent Law as an Equation Example
Product of Powers m n m na a a 2 3 2 3 56 6 6 6 7776
Quotient of Powers ; 0
mm n m n
n
aa a a a
a
3
3 2 3 2 1
2
66 6 6 6 6
6
Product of a Power n
m m na a 3
2 2 3 66 6 6 46656
Power of a Product 2n n na b ab a b
4 4 43 5 3 5 50625
Power of a Quotient n n
n
a a
b b
2 2
2
7 7 49
4 4 16
Complete the table below. The first one is done for you.
Product of Powers m n m na a a 2 3 2 3 54 4 4 4 1024 m n m na a a 21.3 1.3 m n m na a a 2 7
1 1
2 2
m n m na a a 7 9y y
22
Complete the table below. The first one is done for you.
Quotient of Powers m
m n
n
aa
a
6
6 4 6 4 2
4
88 8 8 8 64
8
mm n
n
aa
a
8
8 5
5
1.31.3 1.3
1.3
mm n
n
aa
a
10
10 5
5
1
1 12
2 21
2
mm n
n
aa
a
7
7 9
9
yy y
y
Complete the table below. The first one is done for you.
Product of a Power
n
m m na a 2
4 4 2 88 8 8 16777216
n
m m na a 10
21.3
n
m m na a 3
53
4
n
m m na a 8
11y
Complete the table below. The first one is done for you.
Power of a Product
n n n na b ab a b
2 2 28 3 8 3 64 9 576
n n n na b ab a b
101.3 0.8
n n n na b ab a b
32 16
7 5
n n n na b ab a b
8r s
23
Complete the table below. The first one is done for you.
Power of a Quotient n n
n
a a
b b
3 3
3
8 8 512 or 18.962
3 3 27
n n
n
a a
b b
51.3
0.8
n n
n
a a
b b
23
45
8
n n
n
a a
b b
8r
s
Now let’s use these laws of exponents to simplify expressions which have negative and/or fractional
exponents.
E.g.: Write 2 70.8 0.8 as a single power.
2 72 7 5
5
10.8 0.8 0.8 0.8
0.8
E.g.: Write
3 52 4
4 4
5 5
as a single power.
3 52 4 2 3 4 5 6 20 6 20 14
4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5
E.g.: Write
53
5
1.5
1.5
as a single power.
5
3 3 5 1515 5 10
5 5 5
1.5 1.5 1.51.5 1.5
1.5 1.5 1.5
24
E.g.: Write
5 1
4 4
3
4
9 9
9
as a single power.
5 15 1 4
4 3 14 44 4 44 4 4
3 3 3
4 4 4
9 9 9 99 9
9 9 9
E.g.: Simplify
22
33 3
4 4
2 2 6 8 8 82 83 3 3 3
38
327 27 27 27 27 3 6561
64 64 64 64 64 4 4 65536
E.g.: Simplify
32
3
t
t
32 6
6 3 9
3 3
t tt t
t t
E.g.: Simplify 4 2 2 3m n m n
2 34 2 2 3 4 2 2 3 4 2 6m n m n m m n n m n m n
E.g.: Simplify 4 3
2
6
14
x y
xy
4 3 4 3 3
3 24 1 3 5 3
2 2 5 5
6 6 3 3 3 1 3
14 14 7 7 7 7
x y x y xx y x y x
xy x y y y
E.g.: Simplify 3
4 2 225a b
3 4 3 2 3 12 63 3 3 3
4 2 4 2 3 6 32 1 2 1 2 2 22 2 225 25 25 5 125a b a b a b a b a b
25
E.g.: Simplify 3 1
3 12 2x y x y
3 13 1 3 1 2 2
3 13 1 3 1 2 2 1 22 22 2 2 2 21
1 xx y x y x x y y x y x y x y x
y y
E.g.: Simplify
55 2
1 1
2 2
12
3
x y
x y
5 5
5 11 10 1 5 1 11 6 11 65 5 32 2 532 22 2 2 2 2 2 2 2 2
1 1 1 1 11 11
2 2 2 2 2 2
12 12 1 44 4 4 4 4
33
x y x y yx y x y x y x y y
x y x y x x
E.g.: Simplify
12 4 2
4 7
50
2
x y
x y
11 1 1 112 4 2 4 22 2 2 2
2 3 21 1 34 7 4 7 2 3 2 3
2 32 2 2
50 50 1 1 25 25 525 25
2 2
x y x yx y
x y x y x y x yx y xy
E.g.: Simplify
2
3
2 3
6
x xy
xy
2 2 2 2 2 2 3 2 3 2
3 1 2 3
3 3 3 3 3
22 1 2
2 3 2 3 2 9 18 183
6 6 6 6 6
1 33 3
x xy x x y x x y x y x yx y
xy xy xy xy x y
xx y x
y y
E.g.: Simplify
11 2 1 1
5 4 4 2x y x y
1 1 1
1 1 11 1 12 1 1 2 1 1 1 1 12 2 25 5 104 4 2 4 4 2 4 4 2
71 1 1 21 1 1 2 5 7 71 1 1 3 204 2 4 410 10 4 20 20 20 204 4 2 4
3 3
4 4
1
x y x y x y x y x y x y
xx x y y x y x y x y x
y y
26
E.g.: A cone with equal height and radius has a volume of 318cm . What are the radius and the height to
the nearest tenth of a centimetre?
2
3
r hV
Since the volume is 18 and r h then
2
3
3
3
183
183
3 18 33
54
54
r r
r
r
r
3r
3
3
54
54
2.6cm
r
r
r
The radius and height are about 2.6cm.
Do #’s 3 a, b, d, 4, 5, 6, 7, 8 a c, e, g, 9 a, c, e, g, 10 b, d, f, h, 11, 12, 14, 15 c, d, 16, 17, 19, pp. 241-
243 text in your homework booklet.
Do #’s 1 b, d, 4 a, c, 6 a, c, d, e, f, h, i 9, 10, 11, 12, 14, 17 b, d, 18 b, d, 19, 20, 24, 28 b, c, 29 b, d, 30
32, pp. 246-248 text in your homework booklet.