Question

Cho biểu thức:

\(Q=\left(\dfrac{1}{\sqrt{a}+1}-\dfrac{1}{a+\sqrt{a}}\right):\dfrac{\sqrt{a}-1}{a+2\sqrt{a}+1}\)

a Tìm điều kiện và rút gọn Q

b So sánh Q với 1

Answer 1

a)

ĐKXĐ: \(\left\{{}\begin{matrix}a\ge0\\\sqrt{a}\ne0\\\sqrt{a}-1\ne0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}a>0\\a\ne1\end{matrix}\right.\)

\(M=\left(\dfrac{1}{\sqrt{a}+1}-\dfrac{1}{a+\sqrt{a}}\right)\div\dfrac{\sqrt{a}-1}{a+2\sqrt{a}+1}\)

\(=\dfrac{\sqrt{a}-1}{\sqrt{a}\left(\sqrt{a}+1\right)}\times\dfrac{\left(\sqrt{a}+1\right)^2}{\sqrt{a}-1}\)

\(=\dfrac{\sqrt{a}+1}{\sqrt{a}}\)

b)

M = \(=\dfrac{\sqrt{a}+1}{\sqrt{a}}=1+\dfrac{1}{\sqrt{a}}>1\left(\dfrac{1}{\sqrt{a}}>0\right)\)