Definition

If a is any real number and n is a natural number, then \( { a }^{ n } \) = a x a x a…. n times

where a is called the base, n is called the exponent or index and  \( { a }^{ n } \) is the exponential expression. \( { a }^{ n } \) is read as ‘a raised to the power n’ or ‘a to the power n’ or simply ‘a power n’.

For zero power, we have :

                                            \( { a }^{ 0 } \) = 1 (where a \( \neq \) 0)

For example :

        (i) \( { 7 }^{ 0 } \) =  1    (ii) \( { (-\frac { 2 }{ 3 } ) }^{ 0 } \) = 1    (iii) \(( { \sqrt { 7 }  })^{ 0 } \) = 1

For negative powers, we have :

           \( \sqrt [ n ]{ { a }} \) = \( \frac { 1 }{ { a }^{ n } } \) and \( \frac { 1 }{ { a }^{ -n } } \) = \( { a }^{ n } \)

For example:
    (i) \( { 5 }^{ -2 } \) = \( \frac { 1 }{ { 5 }^{ 2 } } \)
    (ii) \( { -2 }^{ -3 } \) = \( \frac { 1 }{ { -2 }^{ 3 } } \)
    (iii) \( \frac { 1 }{ { 2 }^{ -5 } } \) = \( { 2 }^{ 5 } \)

For fractional indices,  we have :

        \( { \sqrt { a}  }^{ n } \) = \( { a }^{ \frac { 1 }{ n }  } \) and \( \sqrt [ n ]{ { a }^{ m } } \) = \( { a }^{ \frac { m }{ n }  } \)

For example:

    (i) \( { \sqrt { 3 }  }\) = \( { 3 }^{ \frac { 1 }{ 2 }  } \)
    (ii) \( { \sqrt { 8 }  }^{ 3 } \) = \( { 8 }^{ \frac { 1 }{ 3}  } \) 
    (iii) \( \sqrt [ 4 ]{ { 5 }^{ 3 } } \) = \( { 5 }^{ \frac { 3 }{ 4 }  } \)

Finding the value of the Number given in the Exponential Form

Example 1: Find the value of each of the following:

    (i) \( { 12 }^{ 2 } \)    (ii) \( { 8 }^{ 3 } \)    (iii) \( { 4 }^{ 4 } \)

Solution.

(i) We have,

    \( { 12 }^{ 2 } \) = 12 x 12 = 144
(ii) We have,

    \( { 8 }^{ 3 } \) = 8 x 8 x 8

        = (8 x 8) x 8
        = 64 x 8
        = 512
(iii) We Have,

    \( { 4 }^{ 4 } \)= 4 x 4 x 4 x 4

        = (4 x 4 ) x 4 x 4
        = (16 x 4) x 4
        = 64 x 4
        = 256

Example 2: Simplify:

    (i) 2 x \( { 10 }^{ 3 } \)    (ii) \( { 5 }^{ 2 } \) x \( { 4 }^{ 2 } \)    (iii) \( { 3 }^{ 3 } \) x 4

Solution.

(i) We have,

2 x \( { 10 }^{ 3 } \) = 2 x 1000 = 2000     [since \( { 10 }^{ 3 } \)=10 x10 x 10 = 1000]

(ii) We have,

\( { 5 }^{ 2 } \) x \( { 4 }^{ 2 } \)

= 25 x 16 = 400

(iii) We have,

\( { 3 }^{ 3 } \) x 4  = 27 x 4 = 108

Expressing Numbers in Exponential Form

Example 1: Express each of the following in exponential form:

    (i) (-4) x (-4) x (-4) x (-4) x (-4)    (ii) \(\frac{2}{5}\) x \(\frac{2}{5}\) x \(\frac{2}{5}\) x \(\frac{2}{5}\)
Solution. We have,

(i)  (-4) x (-4) x (-4) x (-4) x (-4) = \( { -4}^{ 5 } \)

(ii) \(\frac{2}{5}\) x \(\frac{2}{5}\) x \(\frac{2}{5}\) x \(\frac{2}{5}\) = \( ({ \frac { 2 }{ 5 } ) }^{ 4 } \)

Example 2: Express each of the following in exponential form:

    (i) 3 x 3 x 3 x a x a    (ii) a x a x a x a x a x a x b x b x b c x c x c x c
    (iii) b x b x b x \(\frac{2}{5}\) x \(\frac{2}{5}\)
Solution. We have,

(i) 3 x 3 x 3 x a x a = \( { 3 }^{ 3 } \) x \( { a }^{ 2 } \)

(ii) a x a x a x a x a x a x b x b x b x c x c x c x c = \( { a }^{ 6 } \) x \( { b }^{ 3 } \) x \( { c }^{ 4 } \)

(iii) b x b x b x \(\frac{2}{5}\) x \(\frac{2}{5}\) = \( { a }^{ 3 } \) x \( ({ \frac { 2 }{ 5 } ) }^{ 2 } \)

Example 3: Express each of the following numbers in exponential form:

    (i) 128    (ii) 243    (iii) 3125
Solution.

(i) We have,

        128 = 2 x 2 x 2 x 2 x 2 x 2 x 2
        128 = \( { 2 }^{ 7 } \) 

Express-In-Exponential-Form-example3(i)

(ii) We have,

        243 = 3 x 3 x 3 x 3 x 3
        243 = \( { 3 }^{ 5 } \)

Express-In-Exponential-Form-example3(ii)
(iii) We have,

        625 = 5 x 5 x 5 x 5

        625 = \( { 5 }^{ 4 } \)

Express-In-Exponential-Form-example3(iii)

Positive Integral Exponent of a Rational Number

Let \(\frac{a}{b}\) be any rational number and n be a positive integer. Then,

    \( {(\frac { a } { b })^{ n } } \) = \(\frac{a}{b}\) x \(\frac{a}{b}\) x \(\frac{a}{b}\)…n times
              = \( \frac { a\quad \times \quad a\quad \times \quad a….n\quad times }{ b\quad \times \quad b\quad \times \quad b….n\quad times } \)
              = \( \frac { { a }^{ n } }{ { b }^{ n } } \)
Thus, \( {(\frac { a }{ b })^{ n } } \) = \( \frac { { a }^{ n } }{ { b }^{ n } } \) for every positive integer n.

Example : Evaluate:

        (i) \( {(\frac { 3 } { 7 })^{ 3 } } \)    (ii) \( {(\frac { -2 }{ 5 })^{ 3 } } \)

Solution.

    (i) \( {(\frac { 3 } { 7 })^{ 3 } } \) = \( \frac { { 3 }^{ 3 } }{ { 7 }^{ 3 } } \) = \(\frac{127}{343}\)
   

    (ii) \( {(\frac { -2 } { 5 }) ^{ 3 } } \) = \( \frac { { (-2) }^{ 3 } }{ { 5 }^{ 3 } } \)
                      = \( -\frac { { 2 }^{ 3 } }{ { 5 }^{ 3 } } \)
                      = \(-\frac{8}{125}\)

Negative Integral Exponent of a Rational Number

Let \(\frac{a}{b}\) be any rational number and n be a positive integer.

Then, we define,  \( {(\frac { a }{ b })^{ -n } } \) = \( {(\frac { b }{ a })^{ n } } \)

Example : Evaluate:

        (i) \( {(\frac { 1 } { 2 })^{ -3 } } \)    (ii) \( {(\frac { 2 } { 7 })^{ -2 } } \)

Solution.

    (i) \( {(\frac { 1 }{ 2 })^{ -3 } } \)
     = \( {(\frac { 2 } { 1 })^{ 3 } } \) = \( \frac { { 2}^{ 3 } }{ { 1 }^{ 3 } } \)
     = 8
    

   (ii) \( {(\frac { 2 } { 7 })^{ -2 } } \)
     = \( {(\frac { 7 } { 2 })^{ 2 } } \)

     = \( \frac { { 7}^{ 2 } }{ { 2 }^{ 2 } } \)
     = \(\frac{49}{4}\)

Leave a Reply

Your email address will not be published. Required fields are marked *