1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients. 2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents. 3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

1. 4047 3AM Indices (1) Math Academy®
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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.
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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