In this article of ours, we have discussed Integral Exponents of Rational Numbers. Learn about Positive and Negative Integral Exponents of Rational Numbers in the coming modules. Get to know the Definitions, Solved Examples for understanding the concept better.
Positive Integral Exponent of a Rational Number
Consider a/b as a rational number and n to be a positive integer then
(a/b)n = a/b*a/b*a/b*a/b…… n times
= (a*a*a*a…..n times)/(b*b*b*b….. n times)
= an/bn
Therefore, (a/b)n = an/bn and this is applicable for every positive integer “n”.
Example
Evaluate
(i) (3/4)3
Solution:
= (3/4)3
= 3/4*3/4*3/4
= 3*3*3/4*4*4
= 27/64
(ii) (-2/3)4
= -2/3*-2/3*-2/3*-2/3
= (-2*-2*-2*-2)/3*3*3*3
= 16/81
(iii) (5/4)4
= 5/4*5/4*5/4*5/4
= 5*5*5*5/4*4*4*4
= 625/256
Negative Integral Exponent of a Rational Number
Negative Sign in an Exponent represents the multiplicative inverse or reciprocal. For Negative Exponents, the base shouldn’t be zero as zero doesn’t have a reciprocal. Consider a/b a rational number and n is a positive integer. Then, we can say (a/b)-n = (b/a)n
Example
Evaluate
(i) (3/2)-4
= (2/3)4
= 2/3*2/3*2/3*2/3
= (2*2*2*2)/(3*3*3*3)
= 16/81
(ii) (5)-3
= (1/5)3
=1*1*1/5*5*5
= 1/125
(iii)(1/3)-3
= (3/1)3
= (3)3
= 27