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

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

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

$\underline \green{\textsf{To find:}}[tex]</p><p>[tex]\textsf \blue{The value of}$

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

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

$\textsf \red{Consider,}$

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

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

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

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

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

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

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

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