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Cho \(Q=\frac{2\sqrt{a}+3\sqrt{b}}{\sqrt{ab}+2\sqrt{a}-3\sqrt{b}-6}-\frac{6-\sqrt{ab}}{\sqrt{ab}+2\sqrt{a}+3\sqrt{b}+6}\)
a, Rút gọn Q
B, Chứng minh Q=\(\frac{b+81}{b-81}\)thì \(\frac{b}{a}\)là một số nguyên chia hết cho 3
\(Q=\frac{2\sqrt{a}+3\sqrt{b}}{\sqrt{ab}+2\sqrt{a}-3\sqrt{b}-6}-\frac{6-\sqrt{ab}}{\sqrt{ab}+2\sqrt{a}+3\sqrt{b}+6}\)
\(Q=\frac{2\sqrt{a}+3\sqrt{b}}{\sqrt{a}\left(\sqrt{b}+2\right)-3\left(\sqrt{b}+2\right)}-\frac{6-\sqrt{ab}}{\sqrt{a}\left(\sqrt{b}+2\right)+3\left(\sqrt{b}+2\right)}\)
\(Q=\frac{2\sqrt{a}+3\sqrt{b}}{\left(\sqrt{a}-3\right)\left(\sqrt{b}+2\right)}-\frac{6-\sqrt{ab}}{\left(\sqrt{a}+3\right)\left(\sqrt{b}+2\right)}\)
\(Q=\frac{\left(2\sqrt{a}+3\sqrt{b}\right)\left(\sqrt{a}+3\right)}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)\left(\sqrt{b}+2\right)}-\frac{\left(\sqrt{a}-3\right)\left(6-\sqrt{ab}\right)}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-3\right)\left(\sqrt{b}+2\right)}\)
\(Q=\frac{\left(2\sqrt{a}+3\sqrt{b}\right)\left(\sqrt{a}+3\right)-\left(\sqrt{a}-3\right)\left(6-\sqrt{ab}\right)}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)\left(\sqrt{b}+2\right)}\)
\(Q=\frac{2a+6\sqrt{a}+3\sqrt{ab}+9\sqrt{b}-6\sqrt{a}+a\sqrt{b}+18-3\sqrt{ab}}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)\left(\sqrt{b}+2\right)}\)
\(Q=\frac{2a+9\sqrt{b}+a\sqrt{b}+18}{\left(a-9\right)\left(\sqrt{b}+2\right)}\)
\(Q=\frac{\left(a+9\right)\left(\sqrt{b}+2\right)}{\left(a-9\right)\left(\sqrt{b}+2\right)}=\frac{a+9}{a-9}\)