Question

Rational Exponent
In rational exponent there are positive rational exponent and negative rational exponent.
Positive Rational Exponent:
.
For example:

1. Find (125)2/3   

Solution:

(125)2/3   

125 can be expressed as 5 × 5 × 5 = 5³

So, we have (125)2/3 = (53)2/3 = 53 × 2/3 = 52 = 25



2. Find (8/27)4/3

Solution:

(8/27)4/3

8 = 23 and 27 = 33

So, we have (8/27)4/3 = (23/33)4/3

= [(2/3) 3]4/3

= (2/3)3 × 4/3

= (2/3) 4

= 2/3 × 2/3 × 2/3 × 2/3

= 16/81



3. Find 91/2

Solution:

91/2

= √(2&9)

= [(3)2]1/2

= (3)2 × 1/2

= 3



4. Find 1251/3

Solution:

1251/3

= ∛125

= [(5) 3]1/3

= (5) 3 × 1/3

= 5




Negative Rational Exponent:
We already learnt that if x is a non-zero rational number and m is any positive integer then x-m = 1/xm = (1/x)m, i.e., x-m is the reciprocal of xm.

Same rule exists of rational exponents.

If p/q is a positive rational number and x > 0 is a rational number

Then x-p/q = 1/xp/q = (1/x) p/q, i.e., x-p/q is the reciprocal of xp/q

If x = a/b, then (a/b)-p/q = (b/a)p/q

For example:

1. Find 9-1/2

Solution:

9-1/2

= 1/91/2

= (1/9)1/2

= [(1/3)2]1/2

= (1/3)2 × 1/2

= 1/3


2. Find (27/125)-4/3

Solution:

(27/125)-4/3

= (125/27)4/3

= (53/33)4/3

= [(5/3) 3]4/3

= (5/3)3 × 4/3

= (5/3)4

= (5 × 5 × 5 × 5)/(3 × 3 × 3 × 3)

= 625/81
Integral Exponents of a Rational Numbers
We shall be dealing with the positive and negative integral exponents of a rational numbers.
Positive Integral Exponent of a Rational Number
Let a/b be any rational number and n be a positive integer. Then, 

(a/b)ⁿ = a/b × a/b × a/b × ……. n times 

= (a × a × a ×…….. n times )/( b × b × b ×……….. n times ) 

= aⁿ/bⁿ

Thus (a/b)ⁿ = aⁿ/bⁿ for every positive integer n . 

For example:
Evaluate: 

(i) (3/5)³ 

= 3³/5³ 

= 3 × 3 × 3/5 × 5 × 5

= 27/125

(ii) (-3/4)⁴

= (-3)⁴/4⁴

= 34/44

= 3 × 3 × 3 × 3/4 × 4 × 4 × 4

= 81/256


(iii) (-2/3)⁵

= (-2)⁵/3⁵

= (-2)⁵/3⁵

= -2 × -2 × -2 × -2 × -2/3 × 3 × 3 × 3 × 3

= -32/243


Negative Integral Exponent of a Rational Number
Let a/b be any rational number and n be a positive integer.

Then, we define, (a/b)−n−n = (b/a)ⁿ

For example:

(i) (3/4) −5

= (4/3)⁵
(ii) 4−6

= (4/1) −6

= (1/4)⁶

Also, we define, (a/b) = 1


Evaluate:

(i) (2/3) −3

= (3/2)³

= 3³/2³

= 27/8


(ii) 4−2

= (4/1) −2

= (1/4)²

= 1²/4²

= 1/16


(iii) (1/6) −2

= (6/1)²

= 6²

= 36​

Answer 1

Answer:

let the number be x

(-6)^-1 * x = ( 9 )^-1

= > 1 / 6 * x = 1 / 9

= > x = 1 / 3 x ( - 2 ) / 1

= > - 2 / 3 .

Step-by-step explanation: