Name: ___________________________________

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

15) 5 2 5 2

Distribute First

5 2 5 2

5 4

1

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

Zero Exponents – anything to the zero power is 1

26) Evaluate: 09 27)

0

3

23

6

4

ab

a b

1 1

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