worksheet 3.2 index laws name: - weebly

WorkSHEET 3.2 Index laws Name: ___________________________ Fluency 1 Write 3! in expanded factor form:

3! = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3

2 Write 9" in expanded factor form: 9" = 9 × 9 × 9 × 9

3 Given that 9 is the same as 3 × 3, Write your last answer in terms of the number 3.

9" = 9 × 9 × 9 × 9 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3

4 Take a look at your answer to question 1 and your answer to question 3. What can you say about 3! and 9"?

Clearly: 3! = eight 3’s multiplied together and 9" = eight 3’s multiplied together Without a calculator I don’t know what either answer is, but I do know they are the same value. Hence;

3! = 9"

5 Simplify each of the following (in index form). a 4 ´53 ´ 72 ´ 54 ´ 73

b 5 ´ e2 ´ f 2 ´ 3 ´ e3 ´ f 8 ´ 2 ´ e7 ´ f 5

Answers: a 4 ´ 53 ´ 72 ´ 54 ´ 73

= 4 ´ 53 ´ 54 ´ 72´ 73 = 4 ´ 53+4 ´ 72+3 = 4 ´ 57 ´ 75

c 5 ´ e2 ´ f2 ´ 3 ´ e3 ´ f8 ´ 2 ´ e7 ´ f5 = 5 ´ 3 ´ 2 ´ e2 ´ e3 ´ e7 ´ f2 ´ f8 ´ f5

= 30 ´ e2+3+7 ´ f2+8+5 = 30 e12 f15

WorkSHEET 3.2 Index laws

© John Wiley & Sons Australia, Ltd 2

6 Simplify each of the following. a (2h)5 b (a2b6)3 c (m3n2)4 ´ (m4n2)3

Answers: a (2h)5 = 21´5h1´5 = 25h5 = 32h5

b (a2b6)3 = a2´3b6´3 = a6b18

c (m3n2)4 ´ (m4n2)3 = m3´4n2´4 ´ m4´3n2´3

= m12n8 ´ m12n6 = m12+12 ´ n8+6 = m24n14

7 Simplify each of the following. (a)

(b)

Answers: (a)

(b)

8 Are the following statements true or false? (a) (–2)2 = –4 (b) 32 > 23 (c) 52 < 25

Answers: (a) False (b) True (c) True

9 Is the following statement true or false?

Answer: True

10 Apply first and second index laws to simplify the following expression.

Answer:

11 Simplify each of the following. (a) 2430 (b) 50 × 23 × 34

Answers: (a) 1 (b) 23 × 34

12 Simplify each of the following. (a) (b)

Answers: (a) 7 (b) −4


(b)

Answers: (a) 5

(b)

52432 3210 babba ´´

zwwz

2

64

30425 ´

4 1260a b3 4103z w

138148

1829 qpqqp

=´

226432

373

6254

´´´´yxy

xxyyx11 7

47 6

2424x y x yx y

=

04 6+0 3(2 ) 5-

19 1920 3 (4 3 )´ ÷ ´0

0

6 5 73 4 6´ ´´ ´

54



14 Raise each of the following to the given power. (a) (34)7 (b) (2530)11

Answers: (a) 328 (b) 255

15 Simplify;

4#

2#

13#

(5𝑎𝑏)#

2(5𝑎𝑏)#

Answers;

4# = 1

2# = 1

13# = 1

(5𝑎𝑏)# = 1

2(5𝑎𝑏)# = 2 × (5𝑎𝑏)# = 2 × 1 = 2


(b)

Answers: (a) 66 (b) 334

6 Simplify each of the following.

(a)

(b)

Answers:

(a)

(b)

7 Simplify each of the following, giving your answers in index form. (a) (4j2)4 × (3j3m2)5 (b) 5a0 + 2a3i 2 × 3(ai 3)2

Answers:

(a) 44×35×j23m10 (b) 5 + 6a5i 8

2 3 5 0(6 ) (6 )´6 9

4 5

(3 )(3 )

552

93

2374aaa

aa´´

´

606

7604

43)5(

lklke

´´´

143

3l



8 Simplify each of the following, giving your answers in index form.

(a)

(b)

Answers: (a) 22x19y5

(b)

Simplify each of the following: (a) q10 ÷ q2 (b) d16 ÷ d 2 ´ d5

(c)

(d)

(e)

(f)

(a) q8 (b) d19

(c)

(d)

(e)

(f)

9 If z3 = 5, evaluate each of the following. (a) 7z3 – 4 (b) 2z9 (c) 4z6 + 3z0

Answers: (a) 31 (b) 250 (c) 103

10 At 1 pm, there were two bacteria in a Petri dish. If the number of bacteria doubled every 5 minutes, how many bacteria were in the dish at 2 pm. (use a calculator)

Answer: 8192

11 Expand 310 into factor form Expand 95 into factor form Expand 95 into factor form in Base 3 Show 95 as a base of 3

Answers: 310 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 95 = 9 × 9 × 9 × 9 × 9 95 = 9 × 9 × 9 × 9 × 9 = 32 × 32 × 32 × 32 × 32 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 95 = (32)5

= 310

02233

4532

)(2)2(2

xyxyxyx

´´

4332

242325

)2()3()2()(

mnmnmnm

´´

9 5

3 23 2m n´

3

15

1612aa

10

69

2448ppp ´

256

3128

2121841827cbacba

´´´´

7

5

10

9 675

lk

lk

÷

43 12a

34 5p

29 72 cba

3

4

425lk



Without the aid of a calculator, simplify, leaving your answer in index form:

3$% × 25"

5& × 9"

=3$% × (5%)"

5& × (3%)"

=3$% × 5!

5& × 3!

= 3" × 5'

Without the aid of a calculator, simplify, leaving your answer in index form:

4$# × 7!

49& × 2'

=(2%)$# × 7!

(7%)& × 2'

=2%# × 7!

7( × 2'

= 2$' × 7%

If 𝑎 = 𝑥%, determine 𝑎": 𝑎" = (𝑥%)"

= 𝑥!

If 𝑎 = 2𝑥%, determine 3𝑎%: 3𝑎% = 3 × (2𝑥%)%

= 3 × 2% × 𝑥"

= 12𝑥"

If 𝑎 = 4𝑥%and 𝑏 = 2𝑥& determine %)!

*": 8𝑎&

𝑏" =2 × (4𝑥%)&

(2𝑥&)"

=2 × 4& × 𝑥(

2" × 𝑥$%

=2 × 64 × 𝑥(

16 × 𝑥$%

=8𝑥(



By converting to a common ‘base’ , simplify the following:

Answer:

Simplify each of the following, giving the answer in index form. (a)

(b)

Answers:

(a) 3% × 3& × 2& × 2 × 3 = 3( × 2"

(b) 7%+,&-'+,$,"+ = 7&+,"

162m ´ 8m

23m ´ 44m ( ) ( )( )

2 8

3 4

24 3

43

11

11

16 82 4

2 2

2 2

221

m

m m

m m

mm

m

m

´´

´=

´

=

=

62273 32 ´´´

n

nn

4

532

7777

´´-

worksheet 3.2 index laws name: - weebly

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