Question

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

Answer 1

$\underline{\textsf{Given:}}$

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

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

$\textsf{The value of}$

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

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

$\textsf{Consider,}$

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

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

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

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

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

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

$\underline{\textsf{Answer:}}$

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

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}}}p−3+p−4+p−5+p−6+p−7+p−8+p−9p+p2+p3+p4+p5+p6+p7

\underline{\textsf{To find:}}To find:

\textsf{The value of}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}}}p−3+p−4+p−5+p−6+p−7+p−8+p−9p+p2+p3+p4+p5+p6+p7

\underline{\textsf{Solution:}}Solution:

\textsf{Consider,}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}}}p−3+p−4+p−5+p−6+p−7+p−8+p−9p+p2+p3+p4+p5+p6+p7

\textsf{Multiply both numerator and denominator by}\,\mathsf{p^{10}}Multiply both numerator and denominator byp10

\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}}}=p−3+p−4+p−5+p−6+p−7+p−8+p−9p+p2+p3+p4+p5+p6+p7×p10p10

\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}}=p7+p6+p5+p4+p3+p2+p(p+p2+p3+p4+p5+p6+p7)p10

\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}}=p+p2+p3+p4+p5+p6+p7(p+p2+p3+p4+p5+p6+p7)p10

\mathsf{=p^{10}}=p10

\underline{\textsf{Answer:}}Answer:

\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}}=p^{10}}