For a positive real number and positive integers and

MATHEMATICS · Multi-part · AoPS · Mathematics · Follow the Rules · Follow the Rules

Problem

For a positive real number

a

and positive integers

m, n, p,

and

q

, answer the following. (i) Show, using the definition of rational exponents, that

a^{\frac{m}{n}}=\sqrt[n]{a^{m}}

. (ii) Using the result from part (i), prove that

a^{\frac{m}{n}}\, a^{\frac{p}{q}} = a^{\frac{mq+np}{nq}}.

Hence establish that the law

a^{x}\,a^{y}=a^{x+y}

is valid for rational exponents as well.