Simplify

Algebra Level 3

The value of the following expression: x + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 x − 3 + x − 4 + x − 5 + x − 6 + x − 7 + x − 8 + x − 9 \large\frac{x+x^2+x^3+x^4+x^5+x^6+x^7}{x^{-3}+x^{-4}+x^{-5}+x^{-6}+x^{-7}+x^{-8}+x^{-9}}

x 3 x^3 x − 2 x^{-2} x 10 x^{10} x 3 x^{3} x − 3 x^{-3} x 2 x^2

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

y = x + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 x − 3 + x − 4 + x − 5 + x − 6 + x − 7 + x − 8 + x − 9 = x + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 x − 10 ( x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x ) = x 10 ( x + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 ) x + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 = x 10 \begin{aligned} y & = \frac {x+x^2+x^3+x^4+x^5+x^6+x^7}{x^{-3}+x^{-4}+x^{-5}+x^{-6}+x^{-7}+x^{-8}+x^{-9}} \\ & = \frac {x+x^2+x^3+x^4+x^5+x^6+x^7}{x^{-10}\left(x^7+x^6+x^5+x^4+x^3+x^2+x\right)} \\ & = \frac {x^{10} \cancel {\left(x+x^2+x^3+x^4+x^5+x^6+x^7\right)}}{\cancel{x+x^2+x^3+x^4+x^5+x^6+x^7}} \\ & = \boxed{x^{10}} \end{aligned}

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Hana Wehbi - 3 years, 11 months ago

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