Prove the following result for rational exponents of a positive real number (where…

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Prove the following result for rational exponents of a positive real number

a

(where

p,q,r,s

are integers and

q,s>0

\displaystyle a^{p/q}\;\times\;a^{r/s}=a^{\;\frac{ps+rq}{qs}}

. (i) Express

a^{p/q}

and

a^{r/s}

using radical notation. (ii) Multiply the two radical forms obtained in part (i) and simplify the result so that the new exponent has denominator

qs

. (iii) Conclude that

a^{p/q}\,a^{r/s}=a^{p/q+r/s}

, thereby establishing the general product rule for rational exponents.