1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers. 2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers. 3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.

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) 82 =x
Make the base equal.
(ii) 7=x
e Add ‘lg ’ or ln to both sides
(b) THREE or more terms: 5)3(23 =+ −xx
Substitution
3 Refresh Law of Indices.
Anything to power 0 is 1,
except 0.
3.
0
a = 1
4.
k
a−
= k
a
1
Negative Power
6.
n
b
a






= 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
[A] Two Terms (E Math)
Example 1: Solve the following exponential equations.
(i) ( )
1
2
4 8
x
x
−
= (ii)
2
3x
= x23
27
1
−
Ans: (ii) 3=x
( )
1
( 1)
2 2 3 2(2 ) 2
x
x
−
=
13
2 2 2
(2 ) 2
x
x
−
 
=  
 
3 3
4 2 2
2 2
x
x
−
=
3 3
4
2 2
x x= −
3
2.5
2
x = −
0.6x = −
WS1 – Q1 to 3
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[B] Three Terms (A Math) – Indices as Sum or Difference
Example 2: Solve the equation 8)2(32 11
+= −+ xx
.
Step Split 11
222 =+ xx
8)2(32 11
+= −+ xx
1 2
2 2 3 8
2
x
x  
 = + 
 
Step Replace 2x
with y Let 2x
y =
2 3 8
2
y
y
 
= + 
 
16y =
Step Replace back 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 xx −−+
=+ 12
212 .
Step Split 2 1
2 2 1 2 2x x− −
 + = 
2 1 1
2 2 1
2 2
x
x
 + = 
Step Replace Let 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 = or
1
2
y = −
Step Replace back
1
2
4
x
= or
1
2
2
x
= − (reject as 2 0x
 )
2
2 2
2
x
x
−
=
= −
---------------------------------------------------------------------------------------------------------------------
Example 4: Solve the equation )3(18273 12 xx
=++
. Ans: 1=x
Change 2 x−
to
POSITIVE index
Multiply by 2y
throughout
1
2
3

4. 4047 3AM Exponentials (1)
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in any retrieval system of any nature without prior permission.
Math Academy®
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Example 5: Solve the equation ( )1
3 2 12 3 2x x−
+ = . [4]
Solution: ( )1
3 2 12 3 2x x−
+ =
Step Split ( ) ( )
1
1
23 2 12 3 2 2
x x −
+ = 
Step Replace
Let
1
2
2
x
y = and
2
2x
y =
2 1
3 12 3
2
y y
 
+ =  
 
23
3 12 0
2
y y− − =
4y = or 2y = −
1
2
2 4
x
=
1
2
2 2
x
= − (reject as
1
2
2 0
x
 )
1
22
2 2
x
=
1
2
2
x =
WS 1 Q7-10
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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[D] Indices as a Product
Example 8: Without using calculator, evaluate x
6 , given that 2121
223 +++
= xxx
.
Step Split 2 2
3 3 2 2 2 2x x x
   = 
Step Group terms containing x
3 4 4
2 3 2
x x
x

=

3 4 2
2 3
x
 
= 
 
2
6
3
x
=
Example 9: If xxxx 31223
737x21 = +−
, evaluate
x
2
3
3
7






. Ans:
3
343
Step Split xxxx 31223
737x21 = +−
Step Group terms containing x
Step Group SAME Base
WS 2 Q1-6
DO NOT find x !
Interchangeable
1
2
1
2
3