Mathematics

Problem

(a) Simplify the numerator and the denominator of the fraction

\frac{(9p^{-2}q^{3})^{\tfrac{5}{2}}}{(27p^{4}q^{-6})^{-\tfrac32}}

separately, writing each result with positive exponents. (b) Hence, simplify the entire fraction so that the final answer is expressed as a single product of powers of distinct primes and of the variables

p

and

q

, each with positive exponents only. (c) Find the value of the simplified fraction when

p=\dfrac13

and

q=3

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