so sánh ( $\frac{\pi}{2}$ )^căn 2 và ( $\frac{\pi}{5}$)^-căn3 10/07/2021 Bởi Bella so sánh ( $\frac{\pi}{2}$ )^căn 2 và ( $\frac{\pi}{5}$)^-căn3
Đáp án: $\left(\dfrac{\pi}{5}\right)^{-\sqrt3} > \left(\dfrac{\pi}{2}\right)^{\sqrt2}$ Giải thích các bước giải: Ta có: $\left(\dfrac{\pi}{5}\right)^{-\sqrt3}$ $= \left(\dfrac{5}{\pi}\right)^{\sqrt3}$ Xét $\dfrac{5}{\pi} – \dfrac{\pi}{2}$ $= \dfrac{10 – \pi^2}{2\pi}$ $= \dfrac{(\sqrt{10} – \pi)(\sqrt{10} + \pi)}{2\pi}$ Do $\sqrt{10} > \pi$ nên $\dfrac{(\sqrt{10} – \pi)(\sqrt{10} + \pi)}{2\pi} > 0$ $\to \dfrac{5}{\pi} – \dfrac{\pi}{2} > 0$ $\to \dfrac{5}{\pi} >\dfrac{\pi}{2}$ Ta lại có: $\sqrt3 > \sqrt2$ $\to \left(\dfrac{5}{\pi}\right)^{\sqrt3} > \left(\dfrac{\pi}{2}\right)^{\sqrt2}$ Hay $\left(\dfrac{\pi}{5}\right)^{-\sqrt3} > \left(\dfrac{\pi}{2}\right)^{\sqrt2}$ Bình luận
Đáp án:
$\left(\dfrac{\pi}{5}\right)^{-\sqrt3} > \left(\dfrac{\pi}{2}\right)^{\sqrt2}$
Giải thích các bước giải:
Ta có:
$\left(\dfrac{\pi}{5}\right)^{-\sqrt3}$
$= \left(\dfrac{5}{\pi}\right)^{\sqrt3}$
Xét $\dfrac{5}{\pi} – \dfrac{\pi}{2}$
$= \dfrac{10 – \pi^2}{2\pi}$
$= \dfrac{(\sqrt{10} – \pi)(\sqrt{10} + \pi)}{2\pi}$
Do $\sqrt{10} > \pi$
nên $\dfrac{(\sqrt{10} – \pi)(\sqrt{10} + \pi)}{2\pi} > 0$
$\to \dfrac{5}{\pi} – \dfrac{\pi}{2} > 0$
$\to \dfrac{5}{\pi} >\dfrac{\pi}{2}$
Ta lại có:
$\sqrt3 > \sqrt2$
$\to \left(\dfrac{5}{\pi}\right)^{\sqrt3} > \left(\dfrac{\pi}{2}\right)^{\sqrt2}$
Hay $\left(\dfrac{\pi}{5}\right)^{-\sqrt3} > \left(\dfrac{\pi}{2}\right)^{\sqrt2}$