For positive real numbers and integers

MATHEMATICS · Multi-part · CBSE · Mathematics · Number Systems · Number Systems

Problem

For positive real numbers

a

and integers

m,n>0

, it is asserted that

(a^{m})^{1/n}=a^{m/n}

. Examine the scope of this rule through the tasks below: (i) Prove the rule

(a^{m})^{1/n}=a^{m/n}

for

a>0

by referring to the definition of the principal

n

th root. (ii) Let

a=-8

m=2

, and

n=3

. Compute

(a^{m})^{1/n}

and

a^{m/n}

separately. (iii) Using your results from part (ii), explain why the rule in part (i) fails for negative values of

a

. Identify the step in the proof of part (i) that is not applicable when

a<0