INDICES

When the factors of a product are equal then the product is called a power of the repeated factors.

If n is a positive integer.  The symbol aⁿ, the n^th power of a, is the product of n factors each equal to a

a is called the Base and n the index (plural indices) or exponent.  If the power has a numerical multiplier is called COEFFICIENT.

                                                INDEX

COEFFICIENT             7a³       BASE

 

RULES OF INDICES

1.         m^a × m^b = m^{a+ b} e.g.    

4²×4⁵ = 4⁷ (Multiplication)

2.         m^a ÷ m^b = m^a-b

 e.g.      5⁶÷5⁴ = 5² (Division)

3.         (m^a)^b = m^ab  (Raising to a   power)   

            e.g.       (a²)⁴ = 9² × 9² × 9² × 9² = 9⁸

4.         m^o =1      if m  o (zero index) 6^o = 1

5.         m^-a =  if m  o

Fractional indices

6.         a.        

            b.        

7.         if a^m = aⁿ; then n = m

Example

Simplify

1.     a².a⁷                    

2.    3x⁴.2x⁶     

3.    4x⁷.x³.x⁵  

4.   (z^a. z^b)^d

5.   (x^2a.x^5b)⁴   

6.  (2b⁴)²         

7.    5x⁴÷25x⁷

Solution

1.   a².a⁷ = a²⁺⁷ = a⁹

2.                  3x⁴.2x⁶ = 6x⁴⁺⁶ = 6x¹⁰

3.                  4x⁷.x³.x⁵ = 4x⁷⁺³⁺⁵  = 4x¹⁵

4.                  (z^a.z^b) = (z^a+b)^d = z^d(a+b) = z^ad+bd

5.                  (x^2a.x^5b)⁴ = x^4(2a+5b) = x^8a+20b

6.                  (2b⁴)² = 2²(b⁴)² = 4b⁸

7.                  5x⁴ ÷ 25x⁷ =  x⁴⁻⁷ =  x^-3

 

 

 

Example

Simplify the following

1.)  2.)            3.) 

4.    

Solution

1.            

2.        

3.       

4.        

                             =  4a³

Example

Solve for x in each of the following

^1.                    3^x = 9^x+5

2.                  4^x+1 = 2^x-1

3.                  3^x=(9^2x+3)­(27^1−3x)
Solution

1.                  3^x = 9^x+5

3^x = 3^2(x+5)

3^x = 3^2x+10

Since the base is the some, the index can be equated

            x = 2x + 10

            x = −10

 

2.         4^x+1 = 2^x-1

            2^2(x+1) = 2^x−1

            2^2x+2 = 2^x−1

Equating the powers

            2(x+1) = x – 1

            2x + 2 = x – 1

            2x – x = – 2– 1

                  x = –  3

3.         3^x = (9^2x+3)(27^1−3x)

            3^x = (3^2(2x+3)(3^3(1−3x)

            3^x = (3^4x+6)(3^3-9x)

            3^x = 3^4x+6+3-9x

            3^x = 3^9−5x

Equating the powers

            x = 9 – 5x

            6x = 9

            x =

It is important to note that some exponential equations can be reduced to quadratic form

Example

Solve the following exponential equation

1.                  3^2x – 4 (3^x) + 3 = 0

2.                  5^2x+1 – 26(5^x) + 5 = 0

3.                  3^2x+1 – 28 (3^x-1) + 1 = 0

4.                  2^x + 2^-x = 2

Solution

1.                  3^2x – 4(3^x) + 3 = 0

(3^x)² – 4(3^x) + 3= 0

Let 3^x = b

b² – 4b+3=0

(b −1)(b−3) = 0

b =1 or b = 3              

when b = 1; 3^x = 1

 i.e 3^x = 3⁰

x = 0

or when b = 3;

 3^x = 3¹

 3^x = 3¹ 

 x  = 1

x = 0 or x =1

 

2.         5^2x+1 – 26(5^x) + 5 = 0

            5^2x×5 – 26(5^x)+5 =0

            (5^x)²×5 – 26(5^x) + 5 = 0

            Let  5^x = b

             b²×5 – 26b + 5 = 0

             5b – 26b +5

             (5b – 1) (b – 5) = 0

             when b =

            5^x =  = 5^-1

             x = -1

            or when b = 5

             5^x = 5¹

             x = 1

 x = 1 or x = -1

3.   3^2x+1 – 28(3^x-1) + 1 = 0

      3^2x×3 – 28 (3^x ×3^-1) + 1 = 0

                 

               Let b = 3^x

                b²×3 - 28  + 1 = 0

                3b² -  + 1 = 0

                9b² – 28b + 3 = 0

                (9b – 1) (b-3) = 0

                b =1/9 or b = 3       

          when b = 1/9

         3^x = 1/9 = 3^-2

            x = -2

       or when b = 3

         3^x=  3

         x = 1

         x = -2 or x = 1

 

4.         2^x + 2^-x=2

            2^x +  = 2

          Let 2^x = b

            b +   = 2

            b² + 1 = 2b

            b² – 2b +1 = 0

(b – 1)² = 0

             b = 1 twice

           when 2^x = 1

            2^x = 2⁰

            x = 0 twice

 

Example

Find the value (s) of x for which

 

Solution

Equating the powers since the base is the same

             x²+2 = 3x

            x² – 3x +2 = 0

            (x-2)(x-1) = 0

            x = 2 or x = 1

 

Equating the powers

            x²-2 = 4+5x

            x²-5x-6=0

            (x+6)(x-1)

            x = -6 or x  =1

.Equating the indices

            3x² = 8x+3

            3x² – 8x – 3 = 0

            (3x+1)(x-3) = 0

            x =  or x = 3

 

Example

Solve the following simultaneous equation

1.         3^x-y=27,          3^3x-y = 243

2.         27^4-x = 3^1-y      2^x+1 = 4^-y

 

Solution

1.           3^x-y = 27

                3^3x-y  = 243

          3^x-y = 3³

               x-y = 3 - - - (1)

            3^3x-y = 3⁵

               3x-y = 5 - - - (2)

      Subtract (1) from (2)

            2x = 2

           . x = 1

     Substitute x = 1 into (1)

            y = -2

      x =1 and y = -2

2.         27^4−x = 3^1−y

            2^x+1 = 4^-y

 

 

Solution

            27^4-x = 3^1-y

            3^3(4−x) = 3^1−y

            3^12−3x = 3^1−y

             12-3x = 1-y

            3x – y = 11 - - - (1)

            2^x+1 = 4^−y

            2^x+1 = 2^−2y

            x+1 = - 2y

            x+2y = -1 - - - (2)

            Multiply (1) by 2

                        6x – 2y = 22 - - - (3)

             Add (2) and (3) together

                        7x = 21

                         x = 3

            Substitute x = 3 into (1)

                        y = 9−11

                        y = −2

               x = 3, and y = −2

 

Example

Simplify

1.        

 

Solution

Example

Simplify

           

           

Solution