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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
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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
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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
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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
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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
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* 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
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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
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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
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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