Question

Thực hiện phép nhân :

               \(\dfrac{1}{1-x}.\dfrac{1}{1+x}.\dfrac{1}{1+x^2}.\dfrac{1}{1+x^4}.\dfrac{1}{1+x^8}.\dfrac{1}{1+x^{16}}\)

Answer 1

\(\dfrac{1}{1-x}\cdot\dfrac{1}{1+x}\cdot\dfrac{1}{1+x^2}\cdot\dfrac{1}{1+x^4}\cdot\dfrac{1}{1+x^8}\cdot\dfrac{1}{1+x^{16}}\)

\(=\dfrac{1}{\left(1-x\right)\left(1+x\right)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}\)

\(=\dfrac{1}{\left(1-x^2\right)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}\)

\(=\dfrac{1}{\left(1-x^4\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}\)

\(=\dfrac{1}{\left(1-x^8\right)\left(1+x^8\right)\left(1+x^{16}\right)}\)

\(=\dfrac{1}{\left(1-x^{16}\right)\left(1+x^{16}\right)}\)

\(=\dfrac{1}{1-x^{32}}\)

Answer 2