Rational Exponent

In rational exponent there are positive rational exponent and negative rational exponent.

Positive Rational Exponent:

We know that 2³ = 8. It can also be expressed as 8\(^{\frac{1}{3}}\) = 2. 


In general if x and yare non-zero rational numbers and m is a positive integer such that xᵐ = y then we can also express it as y\(^{\frac{1}{m}}\) = x but we can write y\(^{\frac{1}{m}}\) = \(\sqrt[m]{y}\) and is called mᵗʰ root of y. 


For example, y\(^{\frac{1}{2}}\) = \(\sqrt[2]{y}\), y\(^{\frac{1}{3}}\) = ∛y, y\(^{\frac{1}{4}}\) = ∜y, etc. If x is a positive rational number then for a positive ration exponent p/q we have x₀can be defined in two equivalent form. 

x\(^{\frac{p}{q}}\) = \((x^{p})^{\frac{1}{q}}\) = \(\sqrt[q]{x^{p}}\) is read as qᵗʰ root of xᵖ


x\(^{\frac{p}{q}}\) = \((x^{\frac{1}{q}})^{p}\) = \((\sqrt[q]{x})^{p}\) is read as pᵗʰ power of qᵗʰ root of x 


For example:

1. Find (125)\(^{\frac{2}{3}}\)   

Solution:


(125)\(^{\frac{2}{3}}\)   

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




 Exponents

Exponents

Laws of Exponents

Rational Exponent

Integral Exponents of a Rational Numbers

Solved Examples on Exponents

Practice Test on Exponents


 Exponents - Worksheets

Worksheet on Exponents











8th Grade Math Practice

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