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 81/3 = 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 y1/m = x but we can write y1/m = √(m&y) and is called mth root of y.


For example, y1/2 = √(2&y), y1/3 = ∛y, y1/4 = ∜y, etc. If x is a positive rational number then for a positive ration exponent p/q we have xp/q can be defined in two equivalent form.

xp/q = (xp)1/q = √(q&x)p is read as qth root of xp

xp/q = (x1/q)p = (√(q&x))p is read as pth power of qth root of x


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



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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