1

Unit 2 Roots & Powers

General Outcome: • Develop algebraic reasoning and number sense.

Specific Outcomes:

2.1 Demonstrate an understanding of irrational numbers by:

o representing, identifying and simplifying irrational numbers

o ordering irrational numbers

2.2 Demonstrate an understanding of powers with integral and rational exponents

Topics:

• Perfect Squares & Perfect Cubes (Outcome 2.1) Page 2

• Estimating Roots (Outcome 2.1) Page 8

• Radicals (Outcome 2.1) Page 9

• Mixed & Entire Radicals (Outcome 2.1) Page 11

• Exponent Laws (Outcome 2.2) Page 20

• Negative Exponents (Outcome 2.2) Page 22

• Fractional Exponents (Outcome 2.2) Page 28

• Applying the Exponent Laws (Outcome 2.2) Page 32

• Answers Page 38

2

Unit 2 Roots & Powers

Perfect Squares & Perfect Cubes:

Perfect Squares are numbers that can be expressed as some

number times itself. These are numbers that can be represented

as the area of a square.

Ex) List the first 15 perfect squares that are natural numbers.

Ex) List the first 5 perfect cubes that are natural numbers.

Square Roots & Cube Roots:

Ex) Determine the following without using a calculator.

a) √1296 b) √17283

3

c) √1764 d) √27443

Ex) A cube has a volume of 4913 cubic inches. What is its

surface area?

Ex) A cube has a surface area of 294 cm2. What is its volume?

4

Perfect Squares & Perfect Cubes Assignment:

1) Determine the square root of each of the following.

a) 196 b) 256 c) 289 d) 441 e) 361

2) Determine the cube root of each of the following.

a) 343 b) 512 c) 1000 d) 1331 e) 3375

3) Use factoring to determine whether each number is a perfect square,

a perfect cube, or neither.

a) 225 b) 729 c) 1944

d) 1444 e) 4096 f) 13824

5

4) Determine the side length of each square.

a) b)

Area = 484 2mm Area = 1764 2cm

5) Determine the edge length of a cube with the given volume.

a) volume = 5832 3in b) volume = 15625 3m

6) A cube has a surface area of 6534 square feet. Determine its volume.

6

7) Determine all the perfect square whole numbers between each pair

of numbers.

a) 315 – 390 b) 650 – 750

c) 800 – 925 d) 1200 – 1350

8) a) Write an expression for the surface area of the tent shown below.

(Do not include the floor.)

5

8

x ft

x ft

x ft

x ft

b) Suppose that the surface area of the tent is 90 square feet.

Calculate the value of x.

7

9) Determine the dimensions of a cube for which its surface area is

numerically the same as its volume.

10) Determine the side length of a square with area 4 2121x y .

11) Determine the edge length of a cube with volume 6 364x y

8

Estimating Roots:

Use known perfect squares or perfect cubes as benchmarks to

estimate the value of a root that is not perfect.

Ex) Estimate the value of the following.

a) 8 b) 3 9 c) 13

d) 3 35 e) 34 e) 110

f) 50 g) 3 20 h) 60

9

Radicals:

A radical is a mathematical operation. This involves square

roots, cube roots, forth roots, fifth roots, etc.

Index

n x

Radicand

Even Radicals vs. Odd Radicals

Many radicals result in Irrational Numbers, numbers that

cannot be written as a fraction or non-repeating non-terminating

decimals. If the Radicand is not a perfect square, perfect cube,

etc., then the radical will result in an irrational number.

10

Ex) Without using a calculator, determine whether the

following radicals are Rational or Irrational Numbers.

25 30 38

27

0.4 3 0.125 49

3 16 25

64 0.04

11

Mixed & Entire Radicals:

Radicals are typically reduced and written as mixed radicals for

easier operations.

Ex) Express the following as reduced mixed radicals.

150 72 3 40

Ex) Reduce the following radicals by expressing each as a

mixed radical.

a) 200 b) 80

c) 4 75 d) 3 40

12

e) 63 f) 33 48

g) 11 72 h) 20 20

Ex) Express the following as Entire Radicals.

a) 3 5 b) 4 3

c) 10 12 d) 32 6

13

e) 15 7 f) 16 10

g) 35 4 h) 52 3

Ex) Arrange the following from least to greatest without using

your calculator.

3 7, 4 5, 2 13, 6 2

14

Mixed & Entire Radicals Assignment:

1) Write the following radicals in simplest form.

a) 90 b) 72 c) 5 108

d) 600 e) 7 54 f) 900

g) 4 28 h) 99 i) 6 112

15

2) Write the following radicals in simplest form.

a) 3 16 b) 3 81 c) 3 256

d) 35 128 e) 3 120 f) 33 192

g) 33 135 h) 3 8000 i) 32 500

3) Write each mixed radical as an entire radical.

a) 3 2 b) 4 2 c) 6 5

16

d) 32 2 e) 5 6 f) 35 2

g) 7 8 h) 32 9 i) 14 6

4) Can every mixed radical be expressed as an entire radical? Give examples to

support your answer.

5) Can every entire radical be expressed as a mixed radical? Give examples to

support your answer.

17

6) Express the side length of the square shown below as a radical in simplest

form.

Area = 252 2ft

7) A cube has a volume of 200 2cm . Express its side length as a reduced radical

in simplest form.

8) A square has an area of 54 square inches. Determine the perimeter of the

square as a reduced radical in simplest form.

9) Write each radical in simplest form.

a) 4 48 b) 4 405

c) 5 160 d) 6 2916

18

10) Write each mixed radical as an entire radical.

a) 46 3 b) 47 2

c) 53 4 d) 54 3

11) Shown below is a student’s solution for writing 38 2 as an entire radical.

Identify the error(s) in the work and then write the correct solution.

3 3

3 3

3

3

8 2 8 2

2 2

2 2

4

=

=

=

=

12) Shown below is a student’s solution for simplifying 96 . Identify the error(s)

in the work and then write the correct solution.

96 4 48

2 48

2 8 6

2 4 6

8 6

=

=

=

=

=

19

13) The largest square in this diagram has side length 8 cm. Calculate the side

length and area of each of the two smaller squares. Write the radicals in

simplest form.

20

Exponent Laws:

Review

a bx x =

a bx x =

( )b

ax =

( ) axy =

a

x

y

=

Ex) Simplify the following.

a) ( )( )4 65 4x x b) ( )( )4 6

3

32

8

x x

x

21

c) ( )( )4

5 32x x d) ( )( )

( )

24 5

32

6

3

x x

x

e)

57

3

5

15

x

x

f) ( ) ( )

23 4 4

9 3

2 3

12

xy x y

x y

g) ( )( )( )2

33 25 3a b ab h)

( )

415

5

24

2

x

x

22

Negative Exponents:

Complete the following table

52 42 32 22 12 02 12− 22− 32− 42−

ax− =

Ex) Evaluate the following without using a calculator.

24− 35−

20.5−

32

3

−

a

x

y

−

=

23

Ex) Evaluate the following.

a) 25− b) 34− c) 610−

d)

41

3

−

e)

22

5

−

f) 2

3

4

3

−

−

g) 33

4

−

h) ( )2

3−

− i) ( )3

4−

−

24

Ex) Simplify the following.

a) ( ) ( )4

2 53x x− b) ( )6

4 3x y z−

−

c) ( )3

2 52x y−

− d)

72

5

x y

xy

−

e) 4 3

2

15

35

a b

a b

−

− f) ( ) ( )( )

26

3 4 32 5a b a b−

−

25

Negative Exponents Assignment:

1) Evaluate the powers in each pair without a calculator.

a) 24 and 24− b) 42 and 42−

c) 16 and 16− d) 34 and 34−

2) Write each power using positive exponents.

a) 32− b) 53− c) ( )

27

−−

d)

51

4

−

e) 4

1

6− f)

7

4

3

2

−

−

26

3) Evaluate each power without using a calculator.

a) 23− b) 42− c) ( )5

2−

−

d)

31

3

−

e)

22

3

−−

f) 3

1

5−

4) Use a power with a negative exponent to write an equivalent form for each

number.

a) 1

9 b)

1

5 c) 16

d) 25

9 e) 3 f)

8

27

27

5) Simplify the following. Express each answer with positive exponents.

a) ( )( )3 2 5 62 6x y x y− − b) ( )3

3 25a b−

−

c) 4 3

1

24

8

x y

x y

−

− d) ( )( )6 2 3 4 632 2x y x y− − − −

e) ( )4

33ab−

−− f)

35 1

2

x y

y

−−

28

Fractional Exponents:

Radicals can be represented using exponents that are fractions.

1

2x x= , 1

3 3x x= , 1

4 4x x= , 1

5 5x x=

1

nnx x=

Ex) Evaluate each power

1

327

1

249 ( )1

364−

1

24

9

2

1

38

31

216

( )1

3 29

29

( )m

mnnx x= or

mn x

Ex) Evaluate the following.

a)

2

327 b)

3

225 c)

3

532

d)

7

416−

e)

4

364

−

f)

2

3125

8

−

30

Fractional Exponents Assignment:

1) Evaluate each power without using your calculator.

a) 1

24 b) 1

20.09 c) 1

327−

d) ( )2

364− e)

1

21

16

f) 2

532−

g) 3

29−

h) 3

20.25 i)

3

481

16

j) 1.516−

k)

2

38

27

−

−

l) 0.432−

31

2) Given below is a student’s solution for evaluation a power. Identify any errors

in the solution and then write the correct solution.

5 5

3 3

5

3

5

64 64

125 125

64

125

4

5

1024

3125

−

− =

=

=

=

3) Which is greater 52− or 25− ? Justify your answer.

4) How many times greater is 33 than

53−?

32

Applying the Exponent Laws:

a b a bx x x + =

1a

ax

x

− =

a b a bx x x − = a a

x y

y x

−

=

( )b

a abx x= 1

nnx x=

( ) a a axy x y= ( ) or m

mmnnnx x x=

a a

a

x x

y y

=

Ex) Simplify the following. Express each answer using positive

exponents.

a) ( )( )3 2 2 4x y x y− b) 5 3

2 2

10

2

a b

a b−

c) 4 2 2 3m n m n− d)

4 3

2

16

14

x y

xy

−

33

e) ( )1

3 6 38a b f) 3 1

2 12 2x y x y−

g)

2

2 3

1

2 3

4

2

a b

a b

−

h)

1

2

1

5 2

100

25

a

a b−

i)

1

2

3

2

15

5

a

ab

−

−

− j)

11 3

6 3

8

9 3125

x z

y z

−−

−

−

Applying the Exponent Laws Assignment:

34

1) Simplify the following. Express all answers using positive exponents.

a) 3 4x x− b) 4a a− c) 4 3 2b b b−

d) 5

2

x

x

−

e) 8

3

b

b

−

− f)

4

4

t

t

−

−

g) ( )3

1 2x y−

− − h) ( )

22 22a b

−−

i) ( )3

2 34m n−

j)

4

2 33

2m n

−

− −

35

k) ( )

( )

22 1

33

a b

a b

−−

− l)

( )2

13

2

x y

x y

−−

−

m) 2 4

3 3m m n) 3 1

2 4x x− −

o)

3

4 4

1

2 4

9

3

a b

a b

−− p)

1

36

1

9 2

64x

y z−

−

q) ( )3

4 2 225a b r) 3 1

3 12 2x y x y−

−

36

s)

5

5 2

1 1

2 2

12

3

x y

x y

−

− t)

12 4 2

4 7

50

2

x y

x y

u)

4 13 5

2 4 3

a b c

c a b

− −−

−

v)

( )

( )

21 4 3

22 4

2

4

a b c

a bc

−− −

−

2) Identify any errors in each solution for simplifying an expression and then

write a correct solution.

a)

( )1 1

2 3 1 2 3 12 2

1 3

3

x y x y x x y y

x y

xy

− − − − =

=

=

37

b)

( )

( )( )

( )

43 2

65 5

22 3

30 30

30

m nm n

m n

m n

mn

−−

−−

−=

=

=

c) 1

11 3 1 1 5 12

1 12 2 4 2 4 2

5 11 1

4 2

1 3

4 2

1 3

4 2

1

x y x y x y x y

x y

x y

x y

−− − − −

−

− − −

− −

=

=

=

=

3) If 2x a−= and

2

3y a= , write each expression in terms of a.

a)

221

32x y

b)

33 1

4 2x y−

38

Answers

Perfect Squares & Perfect Cubes Assignment:

1. a) 14 b) 16 c) 17 d) 21 e) 19

2. a) 7 b) 8 c) 10 d) 11 e) 15

3. a) perfect square b) both perfect square & perfect cube c) neither

d) perfect square e) perfect square f) perfect cube

4. a) 22 mm b) 42 cm

5. a) 18 inches b) 25 m

6. 35937 cubic feet

7. a) 324, 361 b) 676, 729 c) 841, 900 d) 1225, 1296

8. a) 245

8

x b) 4 feet

9. 1 1 1

10. 211x y

11. 24x y

Mixed & Entire Radicals Assignment:

1. a) 3 10 b) 6 2 c) 30 3 d) 10 6 e) 21 6 f) 30

g) 8 7 h) 3 11 i) 24 7

2. a) 32 2 b) 33 3 c) 34 4 d) 320 2 e) 32 15 f) 312 3

g) 39 5 h) 20 i) 310 4

3. a) 18 b) 32 c) 180 d) 3 16 e) 150 f) 3 250

g) 392 h) 3 72 i) 1176

6. 6 7 feet

7. 32 25 cm

8. 12 6 inches

9. a) 42 3 b) 43 5 c) 52 5 d) 63 4

10. a) 4 3888 b) 4 4802 c) 5 972 d) 5 3072

13. 1st square: Side Length = 4 2 cm, Area = 32 2cm ;

2nd square: Side Length = cm, Area = 16 2cm

39

Negative Exponents Assignment:

1. a) 16 and 1

16 b) 16 and

1

16 c) 6 and

1

6 d) 64 and

1

64

2. a) 3

1

2 b)

5

1

3 c)

2

1

( 7)− d) 54 e) 46 f)

4

7

2

3

3. a) 1

9 b)

1

16 c)

1

32

− d) 27 e)

9

4 f) 125

4. a) 23− b) 15− c) 4

1

2− d)

23

5

−

e) 1

1

3− f)

33

2

−

5. a) 4

2

12y

x b)

6

9125

b

a c)

5

4

3x

y d)

2

8

4x

y e)

12

4

81b

a f)

9

15

y

x

Exponents Assignment:

1. a) 2 b) 0.3 c) 1

3 d) 16 e)

1

4 f)

1

4 g)

1

27 h) 0.125

i) 27

8 j)

1

64 k)

9

4 l)

1

4

3. 5 22 5− −

4. 83 or 6561

Applying the Exponent Laws Assignment:

1. a) x b) 3

1

a c)

3b d) 7

1

x e)

5

1

b f) 1 g) 3 6x y h)

4

44

a

b

i) 6 9

1

64m n j)

8 1216

81

m n k)

5a

b l)

4

2

y

x m)

2m n) 5

4

1

x

o)

1

2

6

3b

a

− p)

1

2 6

3

4x z

y

− q)

6 3125a b r) 2x

y s)

3

11

2

4y

x

t) 3

2

5

xy

u) 16 3

7

a c

b v)

6 2

564

b c

a 3. a)

5

9

1

a

b) 7

2

1

a