Question

p+p²+p³+p⁴+p⁵+p⁶+p⁷)/(p‐³+p‐⁴+ p‐⁵+p‐⁶+p‐⁷+p‐⁸+p‐⁹)​​

Answer 1

$\bold \red{ \frac{p+p²+p³+p⁴+p⁵+p⁶+p⁷)}{p^{−3}+p^{−4}+p^{−5}+p^{−6}+p^{−7}+p^{−8}+p^{−9}}}$

$\bold \green{ = \frac{p^{49}+p^{48}+p^{47}+p^{46}+p^{45}+p^{44}+p^{43}}{p^{39}+p^{38}+p^{37}+p^{36}+p^{35}+p^{34}+p^{33}}}$

$\bold\pink{ = \frac{p^{16}+p^{15}+p^{14}+p^{13}+p^{12}+p^{11}+p^{10}}{p^6+p^5+p^4+p^3+p^2+p+1} }$

$\bold{ \bold \blue{ =} \blue{\frac{pppppppppp(p^6+p^5+p^4+p^3+p^2+p+1)}{p^6+p^5+p^4+p^3+p^2+p+1}}}$

$\bold \purple{ = p {}^{10}}$

Answer 2

Answer:

Given:

$\mathsf{\dfrac{p+p^2+p^3+p^4+p^5+p^6+p^7}{p^{-3}+p^{-4}+p^{-5}+p^{-6}+p^{-7}+p^{-8}+p^{-9}}}$

$\underline{\textsf{To find:}}$

$\textsf{The value of}$

$\mathsf{\dfrac{p+p^2+p^3+p^4+p^5+p^6+p^7}{p^{-3}+p^{-4}+p^{-5}+p^{-6}+p^{-7}+p^{-8}+p^{-9}}}$

$\red{\underline{\textsf{Solution:}} }$

\Consider,

$\mathsf{\dfrac{p+p^2+p^3+p^4+p^5+p^6+p^7}{p^{-3}+p^{-4}+p^{-5}+p^{-6}+p^{-7}+p^{-8}+p^{-9}}}$

$\textsf{Multiply both numerator and denominator by}\,\mathsf{p^{10}}$

$\mathsf{=\dfrac{p+p^2+p^3+p^4+p^5+p^6+p^7}{p^{-3}+p^{-4}+p^{-5}+p^{-6}+p^{-7}+p^{-8}+p^{-9}}{\times}\dfrac{p^{10}}{p^{10}}}$

$\mathsf{=\dfrac{(p+p^2+p^3+p^4+p^5+p^6+p^7)p^{10}}{p^7+p^6+p^5+p^4+p^3+p^2+p}}$

$\mathsf{=\dfrac{(p+p^2+p^3+p^4+p^5+p^6+p^7)p^{10}}{p+p^2+p^3+p^4+p^5+p^6+p^7}}$

$\mathsf{=p^{10}}$

Hence This is Answer.