Question

Rút gọn biểu thức P= \(\sqrt{1-\frac{1}{2^2}}\)\(\sqrt{1-\frac{1}{3^2}}\)...\(\sqrt{1-\frac{1}{2016^2}}\)

Answer 1

\(=\sqrt{\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)...\left(1-\frac{1}{2016^2}\right)}\)

\(=\sqrt{\frac{\left(2^2-1\right)}{2^2}\frac{\left(3^2-1\right)}{3^2}...\frac{\left(2016^2-1\right)}{2016^2}}\)

\(=\sqrt{\frac{1.3.2.4...2015.2017}{2^2.3^2...2016^2}}=\sqrt{\frac{1.2.3...2015}{2.3.4...2016}.\frac{3.4.5...2017}{2.3.4...2016}}\)

\(=\sqrt{\frac{1}{2016}.\frac{2017}{2}}=\sqrt{\frac{2017}{4032}}\)