EXPONENTIAL FORM
Transcript of EXPONENTIAL FORM
23/09/2017
1
Exponents
EXPONENTIAL FORM
Instead of writing π Γ π Γ π Γ π,
we can write ππ
Terminology:
Factor form
Exponential form
β2π₯7 Power
Coefficient Base
Exponent
Understanding Exponents
ππ = π
EXPONENT LAWS
(2π₯)0 = 0 Everything is
being raised to
the power of 0
40 = 1 Anything to the
power of 0 = 1
2π₯0 = 2 Γ 1 = 2 Only π₯ is being
raised to the
power of 0
ππ Γ ππ = ππ+π
πππ ππ Γ ππππ
= ππππ ππ
Multiply noβs &
then add exponents of the
same bases
ππ . ππ = ππ+π = ππ
Add exponents when multiplying same bases
Multiplying & Dividing
Exponents
EXERCISE
1. 2π₯π¦π§ Γ β3π₯4π¦5π§6
2. 6π₯0 Γ 3
3. β 4π5π Γ β2π6π9 4. 12π₯π¦0 Γ (12π₯π¦)0
ππ Γ· ππ = ππβπ
ππ Γ· ππ = ππβπ = ππ
subtract exponents when dividing same
bases
Multiplying
& Dividing Exponents
23/09/2017
2
EXERCISE
1. 12π₯12π¦9 Γ· 3π₯4π¦3
2. 16π4π6 Γ· β4π4π
3. β24π₯π¦11π§2 Γ· β8π₯π¦π§
4. β16π₯13π¦14 Γ· 6π₯9π¦5
(ππ)π = πππ
ππππ ππ Γ· βππππ = πππ ππ
Divide noβs & then subtract
exponents of the same bases
(ππ )π= ππΓπ = ππ multiply exponents
when a power is raised to a power
(ππ)π= ππππ ππ (π
π)π =
ππ
ππ
(ππ
ππ)π = ππΓπ
ππΓπ =ππ
πππ
(πππ )π= ππΓπ . ππΓπ = 23π₯9 = 8ππ Each factor inside the bracket
gets raised to the power
Raising a
Power to an Exponent
EXERCISE
1. 5π₯4π¦9 2 2. 2π3π6 2 3. β3π₯π¦3 3
4. π12
π10
4
Scientific Notation
1. Move the decimal comma until
after the first non zero digit
2. Write Γ 10β¦.
3. Write down the down the no. of
decimal places moved inβ¦
1. 3 020 000 007 000 , ,
= 3.02 Γ ππππ
Scientific
Notation
23/09/2017
3
EXERCISE
1. 3579 2. πππππ πππ 3. πππ πππ πππ πππ
4. 22
Integers
THE NUMBER LINE
0
ββ; β¦ β π; βπ; π; π; π; β¦ ; +β
2 is bigger than -2 β¦ 2 > -2
-2 is bigger than -1 β¦ -2 < -1
-4 is smaller than 0 β¦ -4 < 0
4 is bigger than 0 β¦ 4 > 0
How to read a number line
-4 + 5 = 1
2-4 = -2
β 5+ 4 = -1
2β 6 = -4
β5 β4 β3 βπ β π π π π
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 β¦ 20
2.2. -20 β¦ -40
2.3. -20 β¦ 0 2.4. 6 β¦ -14
2.5 14 β¦ -6
23/09/2017
4
Multiplying (or dividing) signs:
+ Γ + = +
β Γ β = +
+ Γ β = +
β Γ + = +
SIGNS OF NUMBERS
Same sign β¦ answer POSITIVE
Different sign β¦ answer NEGATIVE
2.3 =6
MULTIPLYING INTEGERS
π Γ βπ = β6
β2 Γ β3 = +6
-2(3) = -6
+2 Γ β3 = β6
+2 Γ +3 = +6
β2 Γ +3 = β6
(-2)(-3) = 6
the different ways of writing multiplication β¦ Dot, Γ & Brackets!
π
π
= 3
DIVIDING INTEGERS
βπ Γ· π = β3
β6 Γ· β2 = +3
π Γ· βπ = β3
β6 Γ· +2 = β3
+6 Γ· +2 = +3
+6 Γ· β2 = β3
βπ
βπ
= 3
= 2 + 3
= 5
ADDING & SUBTRACTING
INTEGERS
= 2 β 3
= -1
2 + (-3)
= 2 β (+3) = 1
(2) - (+3)
2+ (+3)
(3) β (+4)
= 2 - 3 = -1
First multiply the signs and then noβs
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, doesnβt change the
answer
2. COMMUTATIVE PROPERTY
23/09/2017
5
1.1. 3-(-7)
1.2. 4+(-8)
1.3. 6 β(+9)
1.4. 5+(+6)
EXERCISE
1.5. -6Γ -4
1.6. -8Γ· 2
1.7. 30 Γ·(-5)
1.8. 9(-3)
2. Fine the additive inverse of -5.
3. Use the commutative property to make
this expression equal: 20+5=β¦.
4. Use the associative property to make this
expression equal : (6+4)-2β¦
EXERCISE
ππ = 3Γ π = 9
(-π)π = βπ Γ βπ = 16
β’ E.g. 9 = 3 (π ππππ 3 Γ 3)
β16 = πππππππππ (π ππππ β 4 Γ-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. β273
2. 121 6. β81
3. (β4)3 7. 23
4. (β4)2 8. 10003
EXERCISE
Common Fractions
23/09/2017
6
* Common fractions are numbers
that can be written as π
π,
where π β π and are classified as:
TYPES OF FRACTIONS
Improper fractions
Types of
Fractions
Comparing
Fractions
MULTIPLYING FRACTIONS
π
π Γ
π
π =
π Γ π
π Γ π =
ππ
ππ
E.g
1. π
πΓ
π
π=
πΓπ
πΓπ=
ππ
ππ
2. π
βπΓ
βπ
βπ=
πΓβπ
βπΓβπ=
βππ
ππ =
βπ
ππ
1. Multiply numerators 2. Multiply denominators
3. Simplify
Multiplying
Fractions
DIVIDING FRACTIONS π
π Γ·
π
π =
π
πΓ
π
π=
π Γ π
π Γ π=
ππ
ππ
E.g
1. π
πΓ·
π
π=
π
πΓ
π
π=
πΓπ
πΓπ=
ππ
π
Γ7
6
1. Find the reciprocal by βtip& Timesβ 2. Multiply Numerators
3. Multiplying denominators
4. Simplify
Dividing
Fractions
EXERCISE
1. ππ
π Γ
βππ
π
2. βππ
π Γ·
π
βπ
3. ππ
ππΓ·
π
π
4. 1 π
πΓ βπ
π
π
ADDING & SUBTRACTING
FRACTIONS π
π+
π
π=
π + π
π
E.g
2. 9
10β
5
10=
9β5
10=
4
10=
2
5
π
πβ
π
π=
π β π
π or
1. Add or subtract numerators
2. Write down common denominators
3. simplify
Adding &
Subtracting Fractions with
Same Denominator
ADDING & SUBTRACTING
FRACTIONS
π
π+
π
π=
ππ + ππ
ππ
1.Find the LCD
2. Find the numerator πΏπΆπ·
πππππππππ‘ππ Γ ππ’πππππ‘ππ
3. Simplify E.g.
1.
2. π
πβ
π
π=
π π +π(π)
ππ=
ππ+π
ππ=
ππ
ππ
Adding &
Subtracting Fractions with
Different Denominators
23/09/2017
7
EXERCISE
1. 12
16β
9
16
2. β16
5 +
1
2
3. 8
9 +
3
4
4. β4
5β
1
6
SQUARES, SQUARE ROOT
CUBES & CUBE ROOTS IN
FRACTIONS
E.g.
1. π
βπ
π=
ππ
(βπ)π =ππ
ππ
2. ππ
ππ =
ππ
ππ=
ππ
ππ=
π
π
3. (βππ
π
π)π =
βπ
π
π=
βπ
ππ
π= β
ππ
π
1.Square , ,
cube or β the
numerator &
denominator
2. Simplify
Square Roots
of Fractions
EXERCISE
1. ππ
ππ
π 3. π
π
π
π
2.ππ
ππ 4. β
π
π
π
PERCENTAGES
1. Write the % as a
fraction over 100 2. "of " πππππ x
3. Multiply numerators & denominators
4. Simplify
E.g. 1.
ππ
πππΓ
πππ
π
= πππππ
πππ
=R110 Finding Percentages
1. Multiply the fraction
by 100
1 to find the %
2. Multiply numerators
& denominators 3. Simplify
E.g. 2.
ππ
ππΓ
πππ
π
= ππππ
ππ
=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.
Iπππππππ =ππ
πππΓ
ππππ
π
= ππππ
πππ
Total = R300 + R2000
= R2300
Percentage Increase & Decrease
23/09/2017
8
EXERCISE
1. Calculate 25% of R3480
2. Calculate Sallyβs 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. π
π=
π
πΓ
π
π=
π
ππ= π, π
2. π
π=
π
πΓ
ππ
ππ=
ππ
πππ= π, ππ
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 noβs in a
column under each other
2. Fill in zeroβs if need be 3. Add or subtract
Subtracting Decimals Adding Decimals
EXERCISE
1.1. π
π
1.2. π
π
1.3. π
π
1.4 π
ππ
EXERCISE
1. 0.3 + 0.08 + 0.456
2. 3.2 β 1.42
3. 69.07 + 42.3 β 2.813
4. 21 β 3.9 β 0.009
23/09/2017
9
MULTIPLYING DECIMALS
E.g.
1. 0,2 Γ 0,3 =2
10Γ
3
10=
6
100= 0,06
2. 0,49 Γ 3,1 =49
100Γ
31
10=
1519
1000= 1.519
1. Convert decimal to fractions
2. Multiply numerators & denominators
3. Convert back to decimals
DIVIDING DECIMALS
2,4 Γ· 0,64 =24
10Γ·
ππ
πππ
=24
10Γ
100
64
=2400
640
=15
4 Γ 25
25
=375
100
= 3.75
1. Convert
decimal to fractions
2. Divide by βtip & timesβ
3. Multiply
numerators and denominators
4. Convert back to decimal
EXERCISE
1. 6.2 Γ· 0.8
2. 9.5 Γ 0.4
3. 3.3 Γ· 0.1
4. 0.065 Γ 0.22
SQUARES, SQUARE ROOTS, CUBES &
CUBE ROOTS IN DECIMALS
= (ππ
πππ)
=ππ
πππ
= π, πππ
1. Convert decimal to fractions
2. Square ; ππ’ππ ππ β the numerator & the denominator
3. Simplify
4. Convert back to decimal
E.g.
2. π. ππ =25
100
=ππ
πππ
=5
10
= π, π
π. π. π π =π
ππ
π
=ππ
πππ
=ππ
ππππ
= π, πππ
EXERCISE
1. 0,49
2. 0,06 2
3. 0,1253
4. 0,002 3