23/09/2017

1

Exponents

EXPONENTIAL FORM

Instead of writing 𝟑 × 𝟑 × 𝟑 × 𝟑,

we can write 𝟑𝟒

Terminology:

Factor form

Exponential form

−2𝑥7 Power

Coefficient Base

Exponent

Understanding Exponents

𝒂𝟎 = 𝟏

EXPONENT LAWS

(2𝑥)0 = 0 Everything is

being raised to

the power of 0

40 = 1 Anything to the

power of 0 = 1

2𝑥0 = 2 × 1 = 2 Only 𝑥 is being

raised to the

power of 0

𝒂𝒎 × 𝒂𝒏 = 𝒂𝒎+𝒏

𝟒𝒂𝟐 𝒃𝟒 × 𝟔𝒂𝒃𝟓

= 𝟐𝟒𝒂𝟐 𝒃𝟗

Multiply no’s &

then add exponents of the

same bases

𝒙𝟐 . 𝒙𝟑 = 𝒙𝟐+𝟑 = 𝒙𝟓

Add exponents when multiplying same bases

Multiplying & Dividing

Exponents

EXERCISE

1. 2𝑥𝑦𝑧 × −3𝑥4𝑦5𝑧6

2. 6𝑥0 × 3

3. − 4𝑎5𝑏 × −2𝑎6𝑐9 4. 12𝑥𝑦0 × (12𝑥𝑦)0

𝒂𝒎 ÷ 𝒂𝒏 = 𝒂𝒎−𝒏

𝒙𝟗 ÷ 𝒙𝟕 = 𝒙𝟗−𝟕 = 𝒙𝟐

subtract exponents when dividing same

bases

Multiplying

& Dividing Exponents

23/09/2017

2

EXERCISE

1. 12𝑥12𝑦9 ÷ 3𝑥4𝑦3

2. 16𝑎4𝑏6 ÷ −4𝑎4𝑏

3. −24𝑥𝑦11𝑧2 ÷ −8𝑥𝑦𝑧

4. −16𝑥13𝑦14 ÷ 6𝑥9𝑦5

(𝒂𝒎)𝒏 = 𝒂𝒎𝒏

𝟐𝟎𝒂𝟒 𝒃𝟖 ÷ −𝟓𝒂𝒃𝟒 = 𝟒𝒂𝟑 𝒃𝟒

Divide no’s & then subtract

exponents of the same bases

(𝒙𝟒 )𝟑= 𝒙𝟒×𝟐 = 𝒙𝟖 multiply exponents

when a power is raised to a power

(𝒂𝒃)𝒏= 𝒂𝒎𝒃𝒎 𝒐𝒓 (𝒂

𝒃)𝒎 =

𝒂𝒎

𝒃𝒎

(𝒂𝟒

𝒃𝟔)𝟐 = 𝒂𝟒×𝟐

𝒃𝟔×𝟐 =𝒂𝟖

𝒃𝟏𝟐

(𝟐𝒙𝟑 )𝟑= 𝟐𝟏×𝟑 . 𝒙𝟑×𝟑 = 23𝑥9 = 8𝒙𝟗 Each factor inside the bracket

gets raised to the power

Raising a

Power to an Exponent

EXERCISE

1. 5𝑥4𝑦9 2 2. 2𝑎3𝑏6 2 3. −3𝑥𝑦3 3

4. 𝑎12

𝑏10

4

Scientific Notation

1. Move the decimal comma until

after the first non zero digit

2. Write × 10….

3. Write down the down the no. of

decimal places moved in…

1. 3 020 000 007 000 , ,

= 3.02 × 𝟏𝟎𝟏𝟐

Scientific

Notation

23/09/2017

3

EXERCISE

1. 3579 2. 𝟐𝟏𝟗𝟎𝟎 𝟎𝟎𝟎 3. 𝟒𝟑𝟖 𝟎𝟎𝟎 𝟎𝟎𝟎 𝟎𝟎𝟎

4. 22

Integers

THE NUMBER LINE

0

−∞; … − 𝟐; −𝟏; 𝟎; 𝟏; 𝟐; … ; +∞

2 is bigger than -2 … 2 > -2

-2 is bigger than -1 … -2 < -1

-4 is smaller than 0 … -4 < 0

4 is bigger than 0 … 4 > 0

How to read a number line

-4 + 5 = 1

2-4 = -2

– 5+ 4 = -1

2– 6 = -4

−5 −4 −3 −𝟐 − 𝟏 𝟎 𝟏 𝟐

Negative

Numbers

EXERCISE

1.1. 3+5 1.2. 3-5 1.3. 5-3

1.4. -3+5 1.5 -3-5 1.6 12-4+8

1.7. -4-8+ 12

EXERCISE

2.1. -20 … 20

2.2. -20 … -40

2.3. -20 … 0 2.4. 6 … -14

2.5 14 … -6

23/09/2017

4

Multiplying (or dividing) signs:

+ × + = +

− × − = +

+ × − = +

− × + = +

SIGNS OF NUMBERS

Same sign … answer POSITIVE

Different sign … answer NEGATIVE

2.3 =6

MULTIPLYING INTEGERS

𝟐 × −𝟑 = −6

−2 × −3 = +6

-2(3) = -6

+2 × −3 = −6

+2 × +3 = +6

−2 × +3 = −6

(-2)(-3) = 6

the different ways of writing multiplication … Dot, × & Brackets!

𝟔

𝟐

= 3

DIVIDING INTEGERS

−𝟔 ÷ 𝟐 = −3

−6 ÷ −2 = +3

𝟔 ÷ −𝟐 = −3

−6 ÷ +2 = −3

+6 ÷ +2 = +3

+6 ÷ −2 = −3

−𝟔

−𝟐

= 3

= 2 + 3

= 5

ADDING & SUBTRACTING

INTEGERS

= 2 – 3

= -1

2 + (-3)

= 2 – (+3) = 1

(2) - (+3)

2+ (+3)

(3) – (+4)

= 2 - 3 = -1

First multiply the signs and then no’s

Rules of Positive and Negative Numbers

E.g. Additive inverse of 2 is -2 i.e 2+(-2)= 0

Properties of Integers

The order of adding integers does not

matter!

Grouping integers when add and

subtracting, doesn’t change the

answer

2. COMMUTATIVE PROPERTY

23/09/2017

5

1.1. 3-(-7)

1.2. 4+(-8)

1.3. 6 –(+9)

1.4. 5+(+6)

EXERCISE

1.5. -6× -4

1.6. -8÷ 2

1.7. 30 ÷(-5)

1.8. 9(-3)

2. Fine the additive inverse of -5.

3. Use the commutative property to make

this expression equal: 20+5=….

4. Use the associative property to make this

expression equal : (6+4)-2…

EXERCISE

𝟑𝟐 = 3× 𝟑 = 9

(-𝟒)𝟐 = −𝟒 × −𝟒 = 16

• E.g. 9 = 3 (𝑠𝑖𝑛𝑐𝑒 3 × 3)

−16 = 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 (𝑠𝑖𝑛𝑐𝑒 − 4 ×-4 =+16)

e

SQUARING & SQUARE-

ROOTING

Understanding Square-Rooting Squares & Square -Roots

What no. multiplied

by itself three times gives two Q?

CUBING & CUBE-ROOTING

Cubes & Cube-Roots

1. 52 5. −273

2. 121 6. −81

3. (−4)3 7. 23

4. (−4)2 8. 10003

EXERCISE

Common Fractions

23/09/2017

6

* Common fractions are numbers

that can be written as 𝒂

𝒃,

where 𝒃 ≠ 𝟎 and are classified as:

TYPES OF FRACTIONS

Improper fractions

Types of

Fractions

Comparing

Fractions

MULTIPLYING FRACTIONS

𝒂

𝒃 ×

𝒄

𝒅=

𝒂 × 𝒄

𝒃 × 𝒅=

𝒂𝒄

𝒃𝒅

E.g

1. 𝟔

𝟕×

𝟐

𝟑=

𝟔×𝟐

𝟕×𝟑=

𝟏𝟐

𝟐𝟏

2. 𝟑

−𝟒×

−𝟔

−𝟓=

𝟑×−𝟔

−𝟒×−𝟓=

−𝟏𝟖

𝟐𝟎 =

−𝟗

𝟏𝟎

1. Multiply numerators 2. Multiply denominators

3. Simplify

Multiplying

Fractions

DIVIDING FRACTIONS 𝒂

𝒃 ÷

𝒄

𝒅=

𝒂

𝒃×

𝑑

𝑐=

𝒂 × 𝒅

𝒃 × 𝒄=

𝒂𝒅

𝒃𝒄

E.g

1. 𝟔

𝟕÷

𝟏

𝟑=

𝟔

𝟕×

𝟑

𝟏=

𝟔×𝟑

𝟕×𝟏=

𝟏𝟖

𝟕

×7

6

1. Find the reciprocal by “tip& Times” 2. Multiply Numerators

3. Multiplying denominators

4. Simplify

Dividing

Fractions

EXERCISE

1. 𝟏𝟐

𝟕 ×

−𝟏𝟐

𝟓

2. −𝟏𝟔

𝟓 ÷

𝟑

−𝟐

3. 𝟐𝟎

𝟐𝟏÷

𝟒

𝟕

4. 1 𝟑

𝟒× −𝟐

𝟐

𝟑

ADDING & SUBTRACTING

FRACTIONS 𝒂

𝒄+

𝒃

𝒄=

𝒂 + 𝒃

𝒄

E.g

2. 9

10−

5

10=

9−5

10=

4

10=

2

5

𝒂

𝒄−

𝒃

𝒄=

𝒂 − 𝒃

𝒄 or

1. Add or subtract numerators

2. Write down common denominators

3. simplify

Adding &

Subtracting Fractions with

Same Denominator

ADDING & SUBTRACTING

FRACTIONS

𝒂

𝒆+

𝒃

𝒇=

𝒂𝒇 + 𝒃𝒆

𝒆𝒇

1.Find the LCD

2. Find the numerator 𝐿𝐶𝐷

𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 × 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟

3. Simplify E.g.

1.

2. 𝟒

𝟓−

𝟐

𝟑=

𝟒 𝟓 +𝟐(𝟑)

𝟏𝟎=

𝟐𝟎+𝟔

𝟏𝟓=

𝟐𝟔

𝟏𝟓

Adding &

Subtracting Fractions with

Different Denominators

23/09/2017

7

EXERCISE

1. 12

16−

9

16

2. −16

5 +

1

2

3. 8

9 +

3

4

4. −4

5−

1

6

SQUARES, SQUARE ROOT

CUBES & CUBE ROOTS IN

FRACTIONS

E.g.

1. 𝟒

−𝟓

𝟐=

𝟒𝟐

(−𝟓)𝟐 =𝟔𝟒

𝟐𝟓

2. 𝟏𝟗

𝟏𝟎 =

𝟐𝟓

𝟏𝟔=

𝟐𝟓

𝟏𝟔=

𝟓

𝟒

3. (−𝟐𝟕

𝟑

𝟐)𝟑 =

−𝟑

𝟐

𝟑=

−𝟑

𝟐𝟑

𝟑= −

𝟐𝟕

𝟖

1.Square , ,

cube or ∛ the

numerator &

denominator

2. Simplify

Square Roots

of Fractions

EXERCISE

1. 𝟐𝟎

𝟑𝟎

𝟐 3. 𝟑

𝟑

𝟖

𝟑

2.𝟒𝟗

𝟐𝟓 4. −

𝟐

𝟓

𝟑

PERCENTAGES

1. Write the % as a

fraction over 100 2. "of " 𝑚𝑒𝑎𝑛𝑠 x

3. Multiply numerators & denominators

4. Simplify

E.g. 1.

𝟒𝟒

𝟏𝟎𝟎×

𝟐𝟓𝟎

𝟏

= 𝟏𝟏𝟎𝟎𝟎

𝟏𝟎𝟎

=R110 Finding Percentages

1. Multiply the fraction

by 100

1 to find the %

2. Multiply numerators

& denominators 3. Simplify

E.g. 2.

𝟐𝟑

𝟑𝟎×

𝟏𝟎𝟎

𝟏

= 𝟐𝟑𝟎𝟎

𝟑𝟎

=76.67%

What percent is a number?

1. Find the

increase by

calculating the

% of the whole 2. Add the

increase to the

whole 3. Simplify

E.g. 3.

I𝒏𝒄𝒓𝒆𝒂𝒔𝒆 =𝟏𝟓

𝟏𝟎𝟎×

𝟐𝟎𝟎𝟎

𝟏

= 𝟑𝟎𝟎𝟎

𝟏𝟎𝟎

Total = R300 + R2000

= R2300

Percentage Increase & Decrease

23/09/2017

8

EXERCISE

1. Calculate 25% of R3480

2. Calculate Sally’s percentage if she

gets 17 out of 40.

3. Increase R10800 by 20%

4. Decrease R12450 by 16%

Decimal Fractions

CONVERTING A FRACTION TO

A DECIMAL

E.g.1. 𝟏

𝟐=

𝟏

𝟐×

𝟓

𝟓=

𝟓

𝟏𝟎= 𝟎, 𝟓

2. 𝟏

𝟒=

𝟏

𝟒×

𝟐𝟓

𝟐𝟓=

𝟐𝟓

𝟏𝟎𝟎= 𝟎, 𝟐𝟓

1. Multiply the numerator & denominator by the same no - in order

to get the denominator to a power of 10

2. Write in Decimal form

=1

Converting

Fractions to Decimals

ADDING & SUBTRACTING

DECIMALS

E.g. 1.

1. Write the no’s in a

column under each other

2. Fill in zero’s if need be 3. Add or subtract

Subtracting Decimals Adding Decimals

EXERCISE

1.1. 𝟐

𝟓

1.2. 𝟑

𝟒

1.3. 𝟏

𝟖

1.4 𝟕

𝟓𝟎

EXERCISE

1. 0.3 + 0.08 + 0.456

2. 3.2 – 1.42

3. 69.07 + 42.3 − 2.813

4. 21 − 3.9 − 0.009

23/09/2017

9

MULTIPLYING DECIMALS

E.g.

1. 0,2 × 0,3 =2

10×

3

10=

6

100= 0,06

2. 0,49 × 3,1 =49

100×

31

10=

1519

1000= 1.519

1. Convert decimal to fractions

2. Multiply numerators & denominators

3. Convert back to decimals

DIVIDING DECIMALS

2,4 ÷ 0,64 =24

10÷

𝟔𝟒

𝟏𝟎𝟎

=24

10×

100

64

=2400

640

=15

4 × 25

25

=375

100

= 3.75

1. Convert

decimal to fractions

2. Divide by “tip & times”

3. Multiply

numerators and denominators

4. Convert back to decimal

EXERCISE

1. 6.2 ÷ 0.8

2. 9.5 × 0.4

3. 3.3 ÷ 0.1

4. 0.065 × 0.22

SQUARES, SQUARE ROOTS, CUBES &

CUBE ROOTS IN DECIMALS

= (𝟕𝟐

𝟏𝟎𝟐)

=𝟒𝟗

𝟏𝟎𝟎

= 𝟎, 𝟎𝟒𝟗

1. Convert decimal to fractions

2. Square ; 𝑐𝑢𝑏𝑒 𝑜𝑟 ∛ the numerator & the denominator

3. Simplify

4. Convert back to decimal

E.g.

2. 𝟎. 𝟐𝟓 =25

100

=𝟐𝟓

𝟏𝟎𝟎

=5

10

= 𝟎, 𝟓

𝟑. 𝟎. 𝟑 𝟑 =𝟑

𝟏𝟎

𝟑

=𝟑𝟑

𝟏𝟎𝟑

=𝟐𝟕

𝟏𝟎𝟎𝟎

= 𝟎, 𝟎𝟐𝟕

EXERCISE

1. 0,49

2. 0,06 2

3. 0,1253

4. 0,002 3