1. 4047 3AM Indices (1) Math Academy®
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Math Academy®
1
It is not failure itself that holds you back; it is the fear of failure that paralyzes you.
Brian Tracy
Notes: Indices
1 Exponential Equation contains an unknown in the exponent.
2 There are 2 types of equations:
(a) TWO terms: (i) Make the base equal.
(ii) Add ‘ ’ or to both sides
(b) THREE or more terms: Substitution
3 Refresh Law of Indices.
[A] Two Terms (E Math)
Example 1: Solve the following exponential equations.
82 =x
7=x
e lg ln
5)3(23 =+ -xx
Anything to power 0 is 1,
except 0.
3. = 1
0
a
4. =k
a-
k
a
1
Negative Power
6. =
n
b
a
÷
ø
ö
ç
è
æ
n
n
b
a
Same Power
7. mnnm
aa =)(
1. =m n
a a´ nm
a +
2. =nm
aa ÷ nm
a -
Same Base
5. ( )n n n
ab a b= ´
Power x Power

2. 4047 3AM Indices (1) Math Academy®
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Math Academy®
2
(i) (ii) = Ans: (ii)
WS1 – Q1 to 3
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[B] Three Terms (A Math) – Indices as Sum or Difference
Example 2: Solve the equation .
Step Split
Step Replace with Let
Step Replace back
( )
1
2
4 8
x
x
-
=
2
3x
x23
27
1
-
3=x
( )
1
( 1)
2 2 3 2(2 ) 2
x
x
-
=
13
2 2 2
(2 ) 2
x
x
-
æ ö
= ç ÷
è ø
3 3
4 2 2
2 2
x
x
-
=
3 3
4
2 2
x x= -
3
2.5
2
x = -
0.6x = -
8)2(32 11
+= -+ xx
11
222 ×=+ xx
8)2(32 11
+= -+ xx
1 2
2 2 3 8
2
x
x æ ö
× = +ç ÷
è ø
2x
y 2x
y =
2 3 8
2
y
y
æ ö
= +ç ÷
è ø
16y =
2 16x
=
4x =
Change to same base
Equate Powers
Base push up as Numerator( )
( )
1
2
1
3 3
=
=
1
2
3

3. 4047 3AM Exponentials (1)
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stored in any retrieval system of any nature without prior permission.
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Example 3: Solve the equation .
Step Split
Step Replace Let
or
Step Replace back or (reject as )
---------------------------------------------------------------------------------------------------------------------
Example 4: Solve the equation . Ans:
xx --+
=+ 12
212
2 1
2 2 1 2 2x x- -
× + = ×
2 1 1
2 2 1
2 2
x
x
× + = ×
2x
y =
1 1
4 1
2
y
y
+ = ×
1 1
4 1
2
y
y
+ = ×
2
8 2 1y y+ =
2
8 2 1 0y y+ - =
(4 1)(2 1) 0y y- + =
1
4
y =
1
2
y = -
1
2
4
x
=
1
2
2
x
= - 2 0x
>
2
2 2
2
x
x
-
=
= -
)3(18273 12 xx
=++
1=x
Change to
POSITIVE index
2 x-
Multiply by
throughout
2y
1
2
3

4. 4047 3AM Exponentials (1)
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in any retrieval system of any nature without prior permission.
Math Academy®
4
Example 5: Solve the equation . [4]
Solution:
Step Split
Step Replace
Let and
or
(reject as )
WS 1 Q7-10
( )1
3 2 12 3 2x x-
+ =
( )1
3 2 12 3 2x x-
+ =
( ) ( )
1
1
23 2 12 3 2 2
x x -
+ = ´
1
2
2
x
y =
2
2x
y =
2 1
3 12 3
2
y y
æ ö
+ = ´ç ÷
è ø
23
3 12 0
2
y y- - =
4y = 2y = -
1
2
2 4
x
=
1
2
2 2
x
= -
1
2
2 0
x
>
1
22
2 2
x
=
1
2
2
x =
Do not multiply 3(2)=6 !
Different base and different power!
Let the term with the smaller power be y
y 2
y
1
2

5. 4047 3AM Exponentials (1)
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in any retrieval system of any nature without prior permission.
Math Academy®
5
[D] Indices as a Product
Example 8: Without using calculator, evaluate , given that .
Step Split
Step Group terms containing
Example 9: If , evaluate . Ans:
Step Split
Step Group terms containing
Step Group SAME Base
WS 2 Q1-6
x
6 2121
223 +++
=´ xxx
2 2
3 3 2 2 2 2x x x
× ´ × = ×
x 3 4 4
2 3 2
x x
x
´
=
´
3 4 2
2 3
x
´æ ö
=ç ÷
è ø
2
6
3
x
=
xxxx 31223
737x21 ÷= +-
x
2
3
3
7
÷
ø
ö
ç
è
æ 3
343
xxxx 31223
737x21 ÷= +-
x
DO NOT find !x
Interchangeable
1
2
1
2
3