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 = \( { 2 }^{ 7 } \)
(ii) We have,
243 = 3 x 3 x 3 x 3 x 3
243 = \( { 3 }^{ 5 } \)
(iii) We have,
625 = 5 x 5 x 5 x 5
625 = \( { 5 }^{ 4 } \)
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) \( {(\frac { 2 } { 7 })^{ -2 } } \)
= \( {(\frac { 7 } { 2 })^{ 2 } } \)
= \( \frac { { 7}^{ 2 } }{ { 2 }^{ 2 } } \)
= \(\frac{49}{4}\)