Developed by Ms A Simpson

METRO CENTRAL EDUCATION DISTRICT

GRADE 9 MATHEMATICS CONSOLIDATION MODULE: EXPONENTS

GET DIRECTORATE

1

Exponents

Parts of a term

−2𝑥3 power

- 2 is called the coefficient of 𝑥3

𝑥 is called the base or variable and

3 is the exponent of 𝑥 or the highest power of 𝑥

Scientific Notation

A method of expressing numbers that are too large or too small to be written in

decimal form.

NB! " move to left increase,

move to right decreases"

Simplifying with Square roots and Cube roots

Square root is the opposite of the

square 9 = 3 x 3

32 = 3

Cube root- is the opposite of the cube

38 =

32 × 2 × 2

323 = 2

Substitution

determine the Value of a term by replacing each variable with a

quantity.

It is a check to to see if any of the expressions are equal

to each other!

Exponential Equations

notice the "=" sign

ensure that the base is the same on either side before you write

the exponents are equal

Fractions can be rewritten as whole numbers with a negative

exponent

1

𝑎3

⬚= 𝑎−3

RulesLaw 1 - 𝑎𝑛 x 𝑎𝑚 = 𝑎𝑚+𝑛

Law 2 -𝑎𝑚

𝑎𝑛

⬚

= 𝑎𝑚−𝑛

Law 3 - (𝑎𝑚)𝑛 = 𝑎𝑚𝑛

Law 4 - (𝑎𝑏)𝑚 = 𝑎𝑚𝑏𝑚

Law 5 - 𝑎0 = 1; 𝑎 ≠ 1

Law 6 - 𝑎−𝑛 = 1

𝑎𝑛 ; 𝑎 ≠ 0

2

EXPONENTS

INSTRUCTIONS: 1. DO ALL THE EXERCISES IN YOUR CLASSWORK BOOK SHOWING YOUR CALCULATIONS

2. NUMBER ACCORDING TO THE QUESTIONS PER WORKSHEET

3. RATHER DO NOT USE A CALCULATOR AS THIS SERVES AS EXTRA PRACTICE

4. FIRST ATTEMPT TO ANSWER ALL THE QUESTIONS BEFORE YOU CHECK ANSWERS IN THE MEMORANDA

WORKSHEET 1 WORKSHEET 2

QUESTION 1

Evaluate without the use of a calculator

1.1 23 × 22

1.2 23 + 22

1.3 𝑥3 × 𝑥2

1.4 𝑥2 + 𝑥2

1.5 32 × 52

1.6 (5 × 3)2

QUESTION 1

Simplify

1.1 8𝑐𝑏−2

6𝑏𝑐3

1.2. (2𝑡)2(−5𝑡)

(−2𝑡)3

1.3 (2𝑥)2( −1𝑥) (4𝑥)

(2𝑥)3 20

1.4 3𝑎𝑏2 × −4𝑎3𝑏

2𝑎𝑏3× 3𝑎𝑏2

QUESTION 2

Simplify .

2.1 𝑎3 × 𝑎2 × 𝑎5

2.2 3𝑏3 × −2𝑏4 × 𝑏

2.3 −𝑚3 × 𝑚−2 × 𝑚5

QUESTION 2

Simplify

2.1 𝑎3. 𝑏2. 𝑐0

2.2 (−𝑎)2

𝑏3 . 𝑎0

ENJOY!!

3

2.4 (2𝑚)2 × −3𝑚

2𝑚3

2.5 −5𝑚𝑛2 × −3𝑚𝑛

2𝑚3𝑛

2.3 (𝑥. 𝑦)2 .(𝑥. 𝑦0)

2.4 (𝑟)2 (𝑟)0)( −𝑟)

( 𝑟3)2 (𝑟4)−1

2.5 (−𝑚)3 (𝑚)2( −𝑚)

( 𝑚2) ( 𝑚0)

WORKSHEET 3 WORKSHEET 4

QUESTION 1

Simplify and rewrite your answer with a positive exponent i.e. with a

fraction if necessary

1.1 𝑤3. 𝑤−3. 𝑐−2

1.2 (−𝑔)2

𝑔3 . 𝑔2

1.3 (𝑚. 𝑛)2 .(𝑚. 𝑛)−3

1.4 (ℎ2 )(𝑔)3( 𝑓)

(𝑓2𝑔ℎ)−2

1.5 (𝑘𝑙)2 (−𝑘)0(− 𝑙)

( 𝑘)3 (𝑙2)−1

QUESTION 1

Determine the value of each term by substituting

𝒂 = −𝟑, 𝒃 = −𝟏 𝒂𝒏𝒅 𝒏 = 𝟐

1.1 𝑎 × 𝑏𝑛

1.1. 𝑎𝑛 × 𝑏

1.3 (𝑎𝑏)𝑛

1.4. 𝑎𝑛𝑏𝑛

1.5 𝑎3𝑏+𝑛

1.6 𝑎𝑛+2𝑏

QUESTION 2

Simplify

2.1 (𝑐𝑏2)(𝑐2𝑏2)

(𝑏3𝑐2)(𝑏2𝑐3)

QUESTION 2

Simplify.

2.1 (i) √16 + 9 (ii) √16 + √9 (iii) √1296

4

2.2 5𝑎−1𝑐𝑏3

2𝑎2𝑏2𝑐3 × 2𝑎2𝑏2

3𝑎𝑏𝑐3

2.3 −3𝑐𝑥3

23𝑎2𝑥2𝑐3 .

9𝑎2𝑏2𝑐−1

3𝑎𝑏2𝑐3

2.4 3𝑚24𝑛−2

𝑚−2(𝑛)3 .

4 𝑛𝑚3

2(𝑚𝑛)2 2.5.

(2𝑥)24𝑦2

5(𝑥𝑦)−2(𝑥𝑦)3

2.2 (i)√5𝑥2 + 4𝑥2 ; (ii) √𝑥2 + √81𝑤4 ; (iii) √9𝑥2𝑦2

2.3 (i)√126 − 13

; (ii) √1253

+ √13

; (iii) √33753

2.4 (i)√𝑥15𝑦93 ; (ii). √216𝑎6𝑏9 3

; (𝑖𝑖𝑖) √1000𝑎12𝑏183

2.5. (i)√4𝑥12𝑦14 + √𝑥12𝑦14; (ii) √125𝑎6𝑏6 3- √64𝑎6𝑏6 3

WORKSHEET 5 WORKSHEET 6

QUESTION 1

Simplify by writing into scientific notation

1.1 3 427

1.2 354 231

1.3 149

1.4 10 000

1.5 0.00645

1.6 0.105

1.7 0.000984

1.8 0.55

1.9 3 100 × 102

1.10 72 540 000

1.11 0.0982× 10−3

1.12 0.002599 × 103

QUESTION 2

Convert Scientific Notation by writing into decimal form

QUESTION 1

Solve the equations below:

1.1. 8𝑥 = 64

1.2. 4−𝑦 = 16

1.3. 3.4𝑥 = 192

1.4 10𝑥 = 0.0001

1.5 1

3𝑥 =1

27

1.6S 4𝑥−2 = 1

1.7 1

5𝑥 = 25

1.8 2𝑥

5 = 12

4

5

5

2.1 4.55× 10−5

2.2 2.2052 × 103

2.3 725.4 × 10−5

2.4 0. 0408 × 10−3

1.9. The size of a virus is 250 nanometres. 1 nanometre is 1× 109 𝑚.

How many viruses are contained in 1𝑘𝑚.

MEMORANDUM TO EXPONENTS WORKSHEET

WORKSHEET 1 WORKSHEET 2

QUESTION 1

Evaluate without the use of a calculator

1.1 23 × 22 = 23+2 = 25 = 32

1.2 23 + 22 = 8 + 4 = 12 NB! + sign can’t add exponents

1.3 𝑥3 × 𝑥2 = 𝑥3+2 = 𝑥5

1.4 𝑥2 + 𝑥2 = 2𝑥2 like terms

1.5 32 × 52 = (3 × 5)2 = (15)2 = 225 both terms have the same

exponent

so you can multiply and then

apply

the exponent OR

32 × 52 = 9 × 25 = 225

1.6 (5 × 3)2 = (15)2 = 225

QUESTION 1

Simplify

1.1 8𝑐𝑏−2

6𝑏𝑐3 = 4𝑐1−3𝑏−2−1

3

= 4𝑐−2𝑏−3

3 =

4

3𝑐+2𝑏+3

1.2. (2𝑡)2(−5𝑡)

(−2𝑡)3 =

(8𝑡2)(−5𝑡)

−8𝑡3

=(8𝑡2)(−5𝑡)

−8𝑡3 =

−40𝑡3)

−8𝑡3

=5

1.3 (2𝑥)2( −1𝑥) (4𝑥)

(2𝑥)3 20 = (4𝑥2)( −1𝑥) (4𝑥)

(8𝑥3)( 1) 20 = 1

−16𝑥4

2𝑥3 =

−8𝑥1

1 Any fraction with 1in the

= −8𝑥 denominator becomes a

whole number

1.4 3𝑎𝑏2 × −4𝑎3𝑏

2𝑎𝑏3× 3𝑎𝑏2 =

−12𝑎4𝑏3

6𝑎2𝑏5

= −12𝑎4𝑏3

6𝑎2𝑏5 =

−2𝑎2

𝑏3

6

QUESTION 2

Remember the order of multiplication signs, numbers and variables

Simplify .

2.1 𝑎3 × 𝑎2 × 𝑎5 = 𝑎3+2+5 = 𝑎10

2.2 3𝑏3 × −2𝑏4 × 𝑏 = (3 × −2 × 1)( 𝑏3+4+1)

= −6𝑏8

2.3 −𝑚3 × 𝑚−2 × 𝑚5 = (-1 × +1 × +1)( 𝑚3−2+5)

= -1𝑚6

2.4 (2𝑚)2 × −3𝑚

2𝑚3 =

4𝑚2 × −3𝑚

2𝑚3

= −12𝑚3

2𝑚3 = - 6 NB! 𝑚3

𝑚3 = 𝑚3−3

= 𝑚0 = 1

2.5 −5𝑚𝑛2 × −3𝑚𝑛

2𝑚3𝑛 =

+15𝑚2𝑛3

2𝑚3𝑛 Multiply signs, numbers and

add the exponents of the

variables are the same.

= +15𝑚2−3𝑛3−1

2 subtract the exponents of the same

variables or factors in the numerator

= +15𝑚−1𝑛2

2 the factor with the negative exponent

means a whole can be written as

a fraction so it goes to the denominator

=15𝑛2

2𝑚+1

QUESTION 2

Simplify

2.1 𝑎3. 𝑏2. 𝑐0 = 𝑎3𝑏2

2.2 (−𝑎)2

𝑏3 . 𝑎0 =

𝑎2

𝑏3

2.3 (𝑥. 𝑦)2 .(𝑥. 𝑦0) = (𝑥2 𝑦2 ).(𝑥. 1)

=(𝑥3 𝑦2 )

2.4 (𝑟)2 (𝑟)0)( −𝑟)

( 𝑟3)2 (𝑟4)−1 =

(𝑟2 )(1)( −𝑟)

( 𝑟6)( 𝑟−4) NB! ( 𝑟3)2 = ( 𝑟3)(𝑟3) = 𝑟6

also −(𝑟2+1 )

( 𝑟6−4)

= (− 𝑟3 )

( 𝑟2) =

−𝑟

1

= −𝑟

2.5 (−𝑚)3 (𝑚)2( −𝑚)

( 𝑚2) ( 𝑚0) =

(𝑚3 )(𝑚2)( −𝑚)

( 𝑚2) ( 1)

= −𝑚6

𝑚2 = −𝑚4

7

Lets check! 15 𝑚×𝑛×𝑛×𝑚×𝑛

2𝑚×𝑚×𝑚×𝑛 we can now cancel out like we did in Gr. 8

Very hard work if the sum is long

=15 𝑛 × 𝑛

2 𝑚

WORKSHEET 3 WORKSHEET 4

QUESTION 1

Simplify and rewrite your answer with a positive exponent i.e. with a

fraction if necessary

1.1 𝑤3. 𝑤−3. 𝑐−2 = 𝑤0𝑐−2 = 1𝑐−2 or 𝑐−2

= 1

𝑐2

1.2 (−𝑔)2

𝑔3 . 𝑔2 = (−𝑔)2

𝑔3 . 𝑔2

= (𝑔2 )

𝑔5

= 𝑔−3 = 1

𝑔3

1.3 (𝑚. 𝑛)2 .(𝑚. 𝑛)−3 = (𝑚2 𝑛2)(𝑚−3 𝑛−3) = 𝑚−1 𝑛−1)

= 1

𝑚𝑛

QUESTION 1

Determine the value of each term by substituting

𝒂 = −𝟑, 𝒃 = −𝟏 𝒂𝒏𝒅 𝒏 = 𝟐

1.1 𝑎 × 𝑏𝑛 = (−3)(−1)2

= (−3)(−1)(−1) = −3(+1)

= −3

1.2 𝑎𝑛 × 𝑏 = (−3)2(−1)

= (−3)(−3)(−1) = (9)(−1)

= − 9

1.3 (𝑎𝑏)𝑛 = ((−3)(−1))2

= (+3)2 = 9

1.4. 𝑎𝑛𝑏𝑛 = (−3)2(−1)2 = (−3)(−3)(−1)(−1)

= (+9)(+1) = +9

1.5 𝑎3𝑏+𝑛 = (−3)3(−1)+(2)

= (−3)−3+2= (−3)−1

=−1

3

8

1.4 (ℎ2 )(𝑔)3( 𝑓)

(𝑓2𝑔ℎ)−2 = (ℎ2 )(𝑔3)( 𝑓)

(𝑓−2𝑔ℎ)−2 = (ℎ2𝑔3𝑓) (𝑓−2𝑔ℎ)+2

= (ℎ2𝑔3𝑓) (𝑓−2𝑔ℎ)(𝑓−2𝑔ℎ)

= ℎ4𝑔5𝑓−3

= ℎ4𝑔5

𝑓3

1.5 (𝑘𝑙)2 (−𝑘)0(− 𝑙)

( 𝑘)3 (𝑙2)−1 = (𝑘2𝑙2) (1)(− 𝑙)

( 𝑘3 )(𝑙−2)

= −(𝑘2𝑙3 )

( 𝑘3𝑙−2) =

−𝑙

𝑘

1.6 𝑎𝑛+2𝑏 = (−3)(2)+2(−1) = (−3)(2)−2)

= (−3)0 = 1

QUESTION 2

Simplify

2.1 (𝑐𝑏2)(𝑐2𝑏2)

(𝑏3𝑐2)(𝑏2𝑐3) =

(𝑐3𝑏4)

(𝑏5𝑐5)

= 𝑏−1𝑐−2 = 1

𝑏1 𝑐2

2.2 5𝑎−1𝑐𝑏3

2𝑎2𝑏2𝑐3 × 2𝑎2𝑏2

3𝑎𝑏𝑐3 = 5

2 𝑎−1−2𝑏3−2𝑐1−3 ×

2

3 𝑎2−1𝑏2−1𝑐−3

== 5

2 𝑎−3𝑏1𝑐−2 ×

2

3 𝑎1𝑏1𝑐−3

== 5

2 ×

2

3𝑎−3+1𝑏1+1𝑐−2−3 ==

5

3𝑎−2𝑏2𝑐−5

= 5𝑏2

3𝑎2𝑐5

QUESTION 2

Simplify.

2.1 (i) √16 + 9 = √25 = 5

(ii) √16 + √9 = 4 + 3 = 7

(iii) √1296 = √24 . 34 = 22 × 32

= 4 × 9 = 36

2.2 (𝑖)√5𝑥2 + 4𝑥2 = √9𝑥2 = 3𝑥

(ii) √𝑥2 + √81𝑤4 = 𝑥 + 9𝑤2 NB! √

divide exponents by 2

(iii) √9𝑥2𝑦2 = 3𝑥𝑦 √3

divide

exponents by 3

2.3 (i)√126 − 13

= √1253

= 5

(ii) √1253

+ √13

= + 5 + 1 = 6

(iii) √33753

= √33. 533 = 3×5 = 15

9

2.3 −3𝑐𝑥3

23𝑎2𝑥2𝑐3 .

9𝑎2𝑏2𝑐−1

3𝑎𝑏2𝑐3 =

−3

8𝑎−2𝑐1−3𝑥3−2 ×

9

3 𝑎2−1𝑏2−2𝑐−1−3

= −3

8𝑎−2𝑐−2𝑥1 ×

9

3 𝑎1𝑏0𝑐−4

= −3

8×

9

3 𝑎−2+1𝑐−2−4𝑥1𝑏0

= −3

8×

9

3 𝑎−2+1𝑐−2−4𝑥1(1)

= −9

8 𝑎−1𝑐−6𝑥1(1) =

−9𝑥1

8𝑎1𝑐6

2.4 3𝑚24𝑛−2

𝑚−2(𝑛)3 . 4 𝑛𝑚3

2(𝑚𝑛)2 = 3𝑚24𝑛−2

𝑚−2𝑛3 × 4 𝑛𝑚3

2𝑚2𝑛2

= 12

1𝑚2+2𝑛−2−3 ×

4

2𝑚3−2𝑛1−2

= 24

1𝑚4𝑛−5 × 𝑚1𝑛−1

= 24

1𝑚4+1𝑛−5−1 =

24𝑚5

1𝑛6

2.5. (2𝑥)24𝑦2

5(𝑥𝑦)−2(𝑥𝑦)3 = 4𝑥24𝑦2

5𝑥−2𝑦−2𝑥3𝑦3 NB! Do remove brackets first

= 4𝑥24𝑦2

5𝑥−2+3𝑦−2+3 (𝑥𝑦)−2 = (𝑥𝑦)−1(𝑥𝑦)−1

= 4𝑥24𝑦2

5𝑥−2+3𝑦−2+3 = (𝑥−1𝑦−1)(𝑥−1𝑦−1)

= 4𝑥24𝑦2

5𝑥1𝑦1 = (𝑥−2𝑦−2)

= 16𝑥𝑦

5

2.4 (i)√𝑥15𝑦93 = 𝑥5𝑦3

(ii). √216𝑎6𝑏9 3= 6𝑎2𝑏3

(𝑖𝑖𝑖) √1000𝑎12𝑏183 = 10𝑎4𝑏6

2.5. (i)√4𝑥12𝑦14 + √𝑥12𝑦14 = 2𝑥6𝑦7 + 𝑥6𝑦7 = 3𝑥6𝑦7

(ii) √125𝑎6𝑏6 3- √64𝑎6𝑏6 3

= 5 𝑎2𝑏2 − 4𝑎2𝑏2 = 1 𝑎2𝑏2

10

WORKSHEET 5 WORKSHEET 6

QUESTION 1

Simplify by writing into scientific notation

1.1 3 427 = 3.427 × 103

1.2 354 231 =3.54231 × 105

1.3 149 = 1.49 × 102

1.4 10 000 = 1.0 × 104

1.5 0.00645 = 6.45 × 10−3

1.6 0.105 = 1.05× 10−1

1.7 0.000984 = 9.84 × 10−4

1.8 0.55 = 5.5 × 10−1

1.9 3 100 × 102 = 3.1 × 105

1.10 72 540 000 = 7.254 × 107

1.11 0.0982× 10−3 = 9.82 × 10−5

1.12 0.002599 × 103 = 2.599

QUESTION 2

Convert Scientific Notation by writing into decimal form

2.1 4.55× 10−5 = 0.000 0455

2.2 2.2052 × 103 = 2 205.2

2.3 725.4 × 10−5 = 0.00007254

2.4 0. 0408 × 10−3= 0.0000408

QUESTION 1

Solve the equations below:

1.1. 8𝑥 = 64

(23)𝑥 = 26 NB! Prime Numbers

23𝑥= 26 base the same, equate exponents

3𝑥 = 6

∴ 3𝑥 = 6

∴ 𝑥 = 2

1.2 4−𝑦 = 16

(22)−𝑦 = 24

2−2𝑦 = 24 base the same, equate exponents

−2𝑦 = 4

(÷ −2) 𝑦 = −2

∴ 𝑦 = −2

1. 3. 3. 4𝑥 = 192

(÷ 3) (22)𝑥 = 64

22𝑥= 26 base the same, equate exponents

2𝑥 = 6

(÷ 2) ∴ 𝑥 = 3

11

1.4 10𝑥 = 0.0001

10𝑥 = 10−4

∴ 𝑥 = −4

1.5. 1

3𝑥 =

1

27

3−𝑥= 27−1 Any fraction can be written as a

whole 3−𝑥 = (33)−1 number with a changed

exponent sign

3−𝑥 = 3−3

−𝑥 = −3

(÷ −1) ∴ 𝑥 = 3

1.6S 4𝑥−2 = 1 We can change 1 = 40

4𝑥−2 = 40

𝑥 − 2 = 0

𝑥 = 2

1.7 1

5𝑥 = 25

5−𝑥 = 52

−𝑥 = 2

(÷ −1) ∴ 𝑥 = −2

1.8 2𝑥

5 = 12

4

5

2𝑥

5 =

64

5

(× 5) 2𝑥 = 64

1.911 2𝑥 = 26

∴ 𝑥 = 6

12

1.9. The size of a virus is 250 nanometres. 1 nanometre is 1× 109 𝑚.

How many viruses are contained in 1𝑘𝑚.

det1 nanometre = 1000 000 000 𝑚 mine the Value of each if

𝑎 = −3; 𝑏 = −1 ; 𝑛 = 2

1) 1000 000 000 𝑚

250 = 4 000 000 𝑣𝑖𝑟𝑢𝑠𝑒𝑠 𝑖𝑛 𝑎 𝑚𝑒𝑡𝑒𝑟

Det 1 𝑘𝑚 = 1000𝑚 4 000 000 × 1000𝑚 in 1𝑘𝑚

Hence 4.0 × 109 𝑣𝑖𝑟𝑢𝑠𝑒𝑠 in 1𝑘𝑚.