## 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) $${(\frac { 2 } { 7 })^{ -2 } }$$
= $${(\frac { 7 } { 2 })^{ 2 } }$$

= $$\frac { { 7}^{ 2 } }{ { 2 }^{ 2 } }$$
= $$\frac{49}{4}$$