Alg. 2 STEM UNIT II Radical and Exponent Notes · Multiplying, Adding, and Subtracting Radical...
Transcript of Alg. 2 STEM UNIT II Radical and Exponent Notes · Multiplying, Adding, and Subtracting Radical...
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Alg. 2 STEM – UNIT II – Radical and Exponent Notes
Day 1 – Approximating, Evaluating and Simplifying Radicals
Objectives: SWBAT evaluate and simplify radicals (square roots and cube roots)
Perfect Squares: Perfect Cubes:
The product of a polynomial The product of a polynomial multiplied by itself multiplied by itself 3 times
21 = 1 28 = 64 215 = 225 31 = 1 36 = 216
22 = 4 29 = 81 216 = 256 32 = 8 37 = 343
23 = 9 210 = 100 217 = 289 33 = 27 38 = 512
24 = 16 211 = 121 218 = 324 34 = 64 39 = 729
25 = 25 212 = 144 219 = 361 35 = 125 310 = 1000
26 = 36 213 = 169 220 = 400
27 = 49 214 = 196
Evaluating~ Approximate~ Simplify~
Answers is a number Round Combine like terms
Evaluating radicals
1) / \
8
64
8
8
2) / \
6
36
6
6
3) / \
7
49
7
7
4) imaginary
nu r
4
mbe
Can’t take the square
Root of a negative
Number
Because it is a cube root you are looking for trios instead of pairs.
5)
3
/ | \
4
64
4 4
4
6)
3
/ | \
2
8
2 2
2
7)
3
/ | \
3 3
2
3
3
7
8)
3
/ | \
7 7
343
7
7
9)
2
/ \
a
a
a
a
10)
2
4
/ \
b b b b
b • b
b
b
11)
33
/ | \
g
g
g g
g
12)
3 9
3
w
/ | \
wwwwwwwww
w w w
w
Approximate the value of the radical by listing the two integers that the radical lies
between.
13) 18 14) 7 15) 3 36
16 25
4 5
4 5and
4 9
2 3
2 3and
3 327 64
3 4
3 4and
Approximate the radical to the nearest integer.
16) 23 17) 3 72
16 25
4 5
4 5
23 25
4.8
and
is closer to
estimate
3 364 125
4 5
4 5
72 64
4.2
and
is closer to
estimate
Evaluate the following expression if x = 64
18) 5 x 19) 3 x x
5 64
5 8
40
3 64 64
4 64
68
Simplify the following radicals.
20)
/ \
4 3
1
/ \
2
2
2
2 3
21)
/ \
9 2
1
/ \
3
8
3
3 2
22)
/ \
9 8
/ \ / \
3 3 4 2
/ \
2 2
72
3 2 2
6 2
23)
3
/ \
16 3
/ \
4
4
3
8
4
4
b
24)
2
/ \
9 6
/ \
3
6
54
3
3
x
x
25) / \
18
324
18
18
26)
3
3
/ \
8 5
/ | \
2 2
4
2
2 5
0
27)
3
3
7
/ \
8 9
/ | \
2 2
7 y
2
2 9
2
Multiplying, Adding, and Subtracting Radical Expressions
Objectives: SWBAT simplify radicals that are multiplied, added together, and subtracted from each other
Things to remember when multiplying roots….
a b a b
Multiply outside coefficients together
Break down all roots
Simplify the following radical expressions.
1) 6 12 2) 5 20 3) 6 5
6 12 72
72
6 2
5 20 100
100
10
6 5 30
100
4) 32 6 3 3x x 5) 7 4 14 6) 2(4 3)
3 4
2
2
2 6 3 3 6 18
6 3 2
18 2
x x x
x
x
7 4 14 4 98
4 98
4 7 2
28 2
4 3 4 3 16 9
16 9
16 3
48
7) 3 315 25 8) 2 53 34y 12y
3 3 3
3
3
15 25 375
375
5 3
2 5 73 3 3
73
2 3
4 12 36
36
2 4
y y y
y
y y
Adding and Subtracting Radical Expressions
Things to remember when adding and subtracting roots….
You can only add/subtract radicals with the same roots
When adding / subtracting radicals, the coefficients change never the root
If they don’t have like radicals, break them down first to see if you can combine them afterwards
Simplify the following radical expressions.
9) 5 3 8 3 3 10) 12 48
5 3 8 3 1 3
2 3
2 3 4 3
2 3
11) 13 2 7 3 2 12) 6 2 3
13 1 2 7 3 2
combine like terms
13 7 1 2 3 2
6 4 2
Distribute First
6(2 3)
2 6 18
2 6 3 2
13) 3 3250m 54m 14) 2
3 3 4
3 3
3
Breakdown the radicals First
5 2m 3 2m
Only Change Coefficients
2 2m
3 3
3 3
3 3
Distribute First
3 4 3 4
9 8 3 16
No Like Terms to Combine
16 9 8 3
Dividing Radical Expressions
Objectives: SWBAT divide radicals SWBAT rationalize a denominator
Division property of radicals~ a a
b b
You can separate the fraction if that is helpful
Remember to reduce all fractions if possible.
1) 9
4 2)
18
81 3)
32
2 4) 3
1
27 5)
3
3
128
8
9 9
4 4
3
2
Reduce First
18 18 2 2
81 981 9
2
3
Reduce First
32 32 1616
2 12
4
3
33
1 1
27 27
1
3
3
33
3
128 128
8 8
4 2
2
Rationalizing the denominator~ You can’t have a radical in the denominator
After apply the division properties, make sure there isn’t any radicals in the denominator. If there is, multiply the numerator and denominator by the radical.
6) 3
2 7)
5
6 8)
4
7
3 2 3 2
2 2 4
3 2
2
5 6 30
6 6 36
30
6
4 7 28
7 7 49
2 7
7
For the cube roots, we still need to make sure we don’t have a radical in the denominator so we multiply by whatever is necessary.
9) 3
5
2 10) 3
11
4 11)
3
6
3
3 3
3 3 3
3
5 2 2 5 4
2 2 2 8
5 4
2
3 3
33 3
3
11 2 22
4 2 8
4 2 2 so we only need one more two
22
2
3 3
3 3 3
3
6 3 3 6 9
3 3 3 27
6 9
3
12) 64
28 13)
4
4
9 6
3 6 14)
3
3
4 18
5 54 15)
50
5 3
Reduce First
64 16 7 112
28 7 497
4 7
7
4
4
Reduce First
9 6 9
33 6
3
3
3 33
3 3
3 3 3
3 3
Reduce First
4 18 4 18 4 1
5 54 5 35 54
4 3 3 4 9
5 3 3 3 5 27
4 9 4 9
5 3 15
Reduce First
50 5 2 2 3
5 3 5 3 3 3
6 2
39
Conjugates~ a multiple that makes a perfect square
When you have a binomial (two terms) in the denominator, you must multiply the numerator and denominator by the binomial that will make a perfect square. This will cancel out the square root in the denominator.
16) 4
3 1 17)
2
1 6
4 3 1 4 3 4
3 1 3 1 9 1 3 1 3 1
combine like terms
4 3 4 4 3 4
3 1 2
2 can go into each part of the numerator
2 3 22 3 2
1
2 1 6 2 12
1 6 1 6 1 1 6 1 6 36
combine like terms
2 2 3
1 6
2 2 3 2 2 3 or
5 5
Perform Operations with Complex Numbers (Day 1)
Objectives: SWBAT add, subtract, multiply, and divide complex numbers SWBAT solve
equations by taking the square root of a negative number
Imaginary Numbers~
Imaginary Unit~ "𝒊" i=√−𝟏
Complex Numbers~ a real number and an imaginary number put together
Simplify the following square roots.
1) 144 2) 8 3) 196 4) 45 5) 3 64
1 144
12 12
12
/ |\
i
i
1 8
2 2 2
2 2
/ |\
i
i
1 196
14 14
14
/ |\
i
i
1 45
3 3 5
3 5
/ |\
i
i
3 64
3 3 3
3
/ |\
no imaginary
numbers in cube
roots
Solve the following equations using square roots.
Isolate the variable, then take a square root – don’t forget the plus or minus.
6) 𝑥2 = −81 7) 𝑥2 + 15 = 5 8) 2𝑥2 + 11 = −37
2 81
9
x
x i
2
2
2
15 5
10
10
10
x
x
x
x i
2
2
2
2
2 11 37
2 48
24
24
2 6
x
x
x
x
x i
i 1
2i 1
3i i
4i 1
9) 𝑥3 = −27 10) 5𝑥2 + 7 = −9
3 3 3 27
3
x
x
2
2
2
2
5 7 9
5 16
16
5
16
5
4 5
5 5
4 5
5
x
x
x
x
x
x
Add and Subtracting Complex Numbers
Add real numbers together and add imaginary numbers together
Real numbers must always go first in your answer
Simplify the following complex expressions.
11) (7 + 2i) + (2 + 8i) 12) (4 + 3i) – (2 – 8i)
7 2 2 8
9 10
rea l im aginary
i i
i
4 2 3 8
2 11
rea l imaginary
i i
i
13) (7 + 2i) + (7 – 2i) 14) (4 – 5i) – (6 – 2i)
7 7 2 2
14
rea l imaginary
i i
4 6 5 2
2 7
real imaginary
i i
i
Perform Operations with Complex Numbers (Day 2)
Objectives: SWBAT multiply, and divide complex numbers
MULITPLYING COMPLEX NUMBERS
i i =2i i1 = 1 i2 = 1 i3 = 1 or i i4 = 1
Find the following:
The pattern repeats every round of four, so use that to help find the requested
values. Your answer should always be one of the four ( 1 , 1 , 1 ,1 )
1) 16i 2)
37i 3) 52i
16 4 4 4 4
1 1 1 1
1
i i i i i
37 4 4 4 4 4 4 4 4 4 1
1 1 1 1 1 1 1 1 1 1
1
i i i i i i i i i i i
1255 4 3
121 1
1
i i i
i
i
Simplify the following complex expressions.
Combine like terms just like they are variables
4) −6(2 − 3𝑖) 5) (1 − 4𝑖)(2 − 8𝑖)
6(2 3 )
12 18
12 18
i
i
real number always
goes first
i
2
2
2
/
1 4 2 8
2 8 4 32
2 12 32
1
2 12 32 1
2 12 32
30 12
Double Distrib FOIL
i i
i i i
i i
i
i
i
real number always
goes first
i
6) (4 − 𝑖)(3 + 2𝑖) 7) (3 + 2𝑖)(7 + 4𝑖)
2
2
2
/
4 3 2
12 8 3 2
12 5 2
1
12 5 2 1
12 5 2
14 5
Double Distrib FOIL
i i
i i i
i i
i
i
i
real number always
goes first
i
2
2
2
/
3 2 7 4
21 12 14 8
21 26 8
1
21 26 8 1
21 26 8
13 26
Double Distrib FOIL
i i
i i i
i i
i
i
i
real number always
goes first
i
Complex Conjugates~ a complex number that has the same numbers but opposite signs. It is helpful when you have an imaginary number in the denominator
Find the product of the following complex conjugates.
5) (3 + 9𝑖)(3 − 9𝑖) 6) (12 − 5𝑖)(12 + 5𝑖)
2
2
2
/
3 9 3 9
9 27 27 81
9 81
1
9 81 1
9 81
90
Double Distrib FOIL
i i
i i i
i
i
2
2
2
/
12 5 12 5
144 60 60 25
144 25
1
144 25 1
144 25
169
Double Distrib FOIL
i i
i i i
i
i
Simplify each expression.
7) 3 5
2
i
i 8)
5 6
3
i
i
2
2
2
3 5
2
3 5
2
1
3 5 1
2 1
5 3 5 3
2 2 2
i i
i i
i i
i
i
i
i ior
2
2
2
5 6
3
5 6
3
1
5 6 1
3 1
6 5 5 2
3 3
i i
i i
i i
i
i
i
i ior
9) 1
3 2
i
i 10)
7
2 i
2
2
2
2
2
1 3 2
3 2 3 2
/
3 2 3 2
9 6 6 4
3 5 2
9 4
1
3 5 2 1
9 4 1
3 5 2
9 4
1 5 1 5
13 13 13
i i
i i
DoubleDistrib Foil
i i i
i i i
simplify
i i
i
i
i
i
i ior
2
2
2
7 2
2 2
/
7 2 7
4 2 2 2
7 2 7
2
1
7 2 7
2 1
7 2 7
2 1
7 2 7 7 2 7
3 13 13
i
i i
Double Distrib Foil
i
i i
simplify
i
i
i
i
i
i ior
11) 12 6
2 4
i
i
2
2
2
2
2
Re
6 3
1 2
6 3 1 2
1 2 1 2
/
6 12 3 6
1 2 2 4
6 15 6
1 4
1
6 15 6 1
1 4 1
6 15 6
1 4
12 15 12 3
5 5
duce
i
i
i i
i i
Double Distrib Foil
i i i
i i i
simplify
i i
i
i
i
i
ior i
Exponents and Multiplying Monomials
Objectives: SWBAT evaluate and simplify expressions involving real numbers. SWBAT evaluate exponents
1. Monomial ~ a one variable or expression
2. Base ~ the bottom number
3. Coefficient ~ constant or a number before a variable
4. Exponent~ indicates the number of times one multiplies the base by itself
5. Caret~ Indicates that the number after the sign is an exponent
Evaluating powers
1) 34 2)
31
2
3) (–2)4 4) (–5)3 5) –62
3 3 3 3
81
1 1 1
2 2 2
1
8
2 2 2 2
16
5 5 5
125
21 6
1 6 6
36
d) e) Say the following to your neighbor
43 3^ 4
2 2 2
2 2
Rah Ah Roma
Mama Ga Ohh La
Product of Powers:
a b a bx x x
1x x
Multiply all coefficients
Simplify the following expressions. Leave answers in exponential form.
6) 3 5x x 7)
4y y 8) 43 42 9) 3 4x y
3 5
8
x
x
1 4
1 4
5
y y
y
y
3 2
5
4
4
1024
3 4
c '
not the same base
an t combine
x y
10) 3 2 3 4a b a b 11) 3 2n m n 12)
2 39 9 13)
1 3
2 2x x
3 3 2 4
3 3 2 4
3 6
a a b b
a b
a b
3 1 2
3 1 2
4 2
n n m
n m
n m
2 3
5
9
9
59,049
1 3
2 2
4
2
2
x
x
x
Multiply the following expressions. Leave answers in exponential form.
14) 6 28 4m m 15) 3 26 p p 16) 2 78 8
6 2
6 2
8
8 4
32
36
m m
m
m
3 2
3 2
5
6
6
6
p p
p
p
2 7
9
8
8
134,217,728
17) 4 52 3
3 4
x x 18) 1 5
3 34 2
m m 19) 3 7 a a
4 5
4 5
9
2 3
3 4
6
12
1
2
x x
x
x
1 5
3 3
1 5
3 3
6
3
2
4 2
8
8
8
m m
m
m
m
2
3 7
21
a a
a
19) 3 b a b 20) 32 c d 21) 2 34 3 j jk
3
3 1 1
3 2
1 1
1
a b b
a b
a b
3
3
2 1
2
c d
c d
2 3
2 1 3
3 3
4 3
12
12
j j k
j k
j k
22) 32 5 ab b ac 23) 2 62 2
3
1 1 1 4
2 5
2 1 5
10
10
a a b b c
a b c
a b c
2 6
2 6
8
1 2 2
1 2
1 2
1 256
256
Quotient, Powers, Negative, and Zero Exponents
Objectives: SWBAT evaluate expressions with exponents
Power of a product property
1. b
a a bx x
2. Distribute to all parts inside the ( )
3. Remember the order of operations, then combine like terms by
adding exponents
Simplify the following monomials.
1) 2
32 2) 3
4x 3) 5
3x
y 4) 4y
3 2
6
2
2
64
4 3
12
x
x
3 5
15
x
x
y
y
4y
5) 2
3 2a b 6) 8
7m n 7) 2
54 3x 8) 2 3
6 5 23y x y z
3 2 2 2
6 4
a b
a b
87
8 7 8 1 8
56 8
1
1
1
m n
m n
m n
2 5 2
10
10
4 3
4 9
36
x
x
x
32 6 2 5 3 2 3 1 3
12 15 6 3
15 6 12 3
15 18 3
15 18 3
3 1
9 1
9 1
9
9
y x y z
y x y z
x y y z
x y z
x y z
Quotient of a power property
1.
mm n
n
xx
x
2. Reduce the confidents like fractions
Simplify the following monomials.
9) 4y
y 10)
9
5
7
7 11)
7
3
8
2
m
m 12)
6 4
5 6
3
9
w x
w x
4 1
3
y
y
9 5
4
7
7
2401
7 3
4
8
2
4
m
m
6 4
5 6
6 5
6 4
2
3
9
1 1
3 1
3
w x
w x
w
x
w
x
13)
34
2
x
y 14)
42
5
4
12
m
n 15)
2
3
23
6
4
ab
a b
4 3
2 3
12
6
x
y
x
y
42
5
4 2 4
4 5 4
8
20
Re
( )
1
3
1
3
81
duce inside
first
m
n
m
n
m
n
23
2 3 2 1 2
23
6 2
23
6 2
23 2
6 1
2
5
2 1 2
2 5 2
2
10
6
4
6
16
Re
( )
6
16
3 1
8 1
3
8
3
8
9
64
ab
a b
ab
a b
duce inside
first
a b
a b
b
a
b
a
b
a
b
a
Negative Exponents
1. 1m
mx
x
2. 1 n
nx
x
3. Can’t have a negative exponent in your answer
Simplify the following expressions and write answers without negative exponents.
17) 32 18) 5 93 3
19) 9
11
5
5
20) 2
1
6
3
1
2
1
8
5
9
9 5
4
3
3
1
3
1
3
1
81
11 9
11 9
20
1
5 5
1
5
1
5
26
36
21) 3 8x x 22) 3 2 4ab a b 23)
5
2 3
3
6
m n
m n
24)
4 3 4
2 5 4
12
15
w x z
w x z
3
8
8 3
5
1
1
x
x
x
x
2 3 4
31 3
4
4
4 3
4
1
a a b b
ba
b
ab
a
b
5 1
2 3
5 2
3 1
5 2
3 1
7
2
7
2
3
6
1 1
2 1
1 1
2 1
1 1
2 1
2
m n
m n
m m
n
m
n
m
n
m
n
4 3 4
2 5 4
3 50
2 4
3 5
2 4
8
6
12
15
4 1
5 1
4 11
5 1
4
5
w x z
w x z
x xz
w w
x
w
x
w
Rational Exponents (Day 1)
Objectives: SWBAT evaluate expressions with exponents
Rational (Fraction) Exponents
1.
ab abx x
Rewrite in radical form and simplify.
1) 1
29 2) 1
2225 3) 1
327 4) 1
24 5) 1
2( 64)
19
9
3
1225
225
15
13
3
27
27
3
14
4
2i
164
64
8i
6) 1
6 38y 7) 1
416 8) 1
220 9) 1
3 232x 10) 1
316
63
2
8
2
y
y
14 16
2
120
2 5
332
4 2
x
x x
13
3
16
2 2
Almost always do the root before the power
11) 3
225 12) 5
24 13) 4
3 38w 14) 2
327
3
3
25
5
125
5
5
4
2
32
433
4
4
8
2
16
w
w
w
2
3
2
27
3
9
15) 3
22 16) 2
4.5 34x 17) 4
23 18) 3
24
33
3 3 3
3 3
29 32
2 2 9
3 3 2
2
33
23 3
3 3
4
4
4
4
2 2
x
x
x
x
x
2
Re
3
9
duce
3
3
3
4
2
8
8
i
i
i
Rewrite in exponential form.
19) 5
6 20) 83 7 21) 8 22) 3 9
5
26 8
37 1
28 1
39
Operations with Rational Exponents – Putting it all together
Objectives: SWBAT evaluate expressions with exponents
Simplify. Write your answers in exponential form AND radical from.
1) 1 3
5 52 2 2)
41
67
3)
5
8
1
8
3
3
1 3
5 5
4
5
5 4
2
E : 2
R : 2
14
6
4
6
2
3
3 2
7
7
E : 7
R : 7
5 1
8 8
4
8
1
2
3
3
E : 3
R : 3
4) 1
4 10 236m n 5) 2
4
3
5
5
6)
18 12 3
2 9
27x y
x y
1 1 1
2 2 24 10
2 5
2 5
2 5
36 m n
6m n
E : 6m n
R : 6m n
42
3
2
3
2
3
3 2
5
5
E : 5
R : 5
18 6 12 9 3
12 3 3
1 1 1
3 3 3
1 1
3 3
1 1
3 3
3
27
27
27
3
: 3
: 3
Reduce Inside
First
x y
x y
x y
x y
E x y
R xy
7)
21 326
8) 4 7
6 3 5
42x z
6x y z 9)
2 5
7 1410 10
1 2
2 3
1
3
1
3
3
6
6
E : 6
R : 6
3
2
3
2
3
2
7
7:
7:
Reduce
First
y z
x
y zE
x
y zR
x
2 5
7 14
5
49
5
49
49 5
10
10
E :10
R : 10
10) 32 2 11) 1 1
3 32 4 12) 3x 364 2
3
11
32
1 1
2 3
5
6
5
6
6 5
2 2
2 2
2
2
E : 2
R : 2
2
1 123 3
1 2
3 3
1 2
3 3
Convert 4 2
2 2
2 2
2
E : 2
R : 2
3 3
6 3 3
64 2
2 2
6 3 3
9 3
3
x
x
x
x
x