Question

if p and q are positive integers such that p^q=q^p and q=27p, then find value of p^13×q^1/9.

Answer 1

so, the value of p^13 × q^1/9 = $\sqrt[9]{3^{118}\times2^2}$

two positive integers p and q in such a way that p^q = q^p and q = 27p

so, p^q = (27p)^p

⇒p^q = 27^p p^p

⇒p^q × p^p = (3)^(3p)

⇒p^(q - p) = (3)^(3p)

on comparing both sides,

p = 3 and q - p = 3p

⇒q = 4p = 12

so, p^13 × q^(1/9)

= 3^13 × (12)^1/9

= 3^13 × 3^1/9 × (4)^1/9

= 3^(13 + 1/9) × (2)^2/9

= 3^(118)/9 × 2^2/9

= $\sqrt[9]{3^{118}\times2^2}$

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Answer 2

Answer:

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