Mathematics

Problem

A student begins to show that

(ab)^{m/n}=a^{m/n}\,b^{m/n}

for

a,b>0

and integers

m,n>0

. Some steps are omitted. Complete the proof as guided below: a. (i) Starting from

(ab)^{m/n}=\sqrt[n]{(ab)^{m}}

, express

(ab)^{m}

as a product involving

a^{m}

and

b^{m}

. b. (ii) Continue the working to arrive at

(ab)^{m/n}=a^{m/n}\,b^{m/n}

, giving reasons for each step used. c. (iii) State the conditions on

a

and

b

that ensure the result is valid.