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\(A=\frac{9a^5-ab^4-18a^4b+2b^5}{3a^2b^2+ab^4-6a^2b^3-2b^5}\)
\(=\frac{a\left(9a^4-b^4\right)-2b\left(9a^4-b^4\right)}{ab^2\left(3a^2+b^2\right)-2b^3\left(3a^2+b^2\right)}\)
\(=\frac{\left(9a^4-b^4\right)\left(a-2b\right)}{\left(3a^2+b^2\right)\left(ab^2-2b^3\right)}\)
\(=\frac{\left(3a^2-b^2\right)\left(3a^2+b^2\right)\left(a-2b\right)}{\left(3a^2+b^2\right)b^2\left(a-2b\right)}\)
\(=\frac{3a^2-b^2}{b^2}\)
\(=3.\left(\frac{a}{b}\right)^2-1=3.\left(\frac{2}{3}\right)^2-1=\frac{1}{3}\)
Ta có:
\(4a^2+b^2=5ab\Leftrightarrow4a^2+b^2-4ab-ab=0\)
\(\Leftrightarrow4a\left(a-b\right)-b\left(a-b\right)=0\)
\(\Leftrightarrow\left(a-b\right)\left(4a-b\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a-b=0\\4a-b=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}a=b\left(ktm\right)\\4a=b\left(tm\right)\end{matrix}\right.\)
\(\Rightarrow4a=b\)
\(\Rightarrow\dfrac{5ab}{3a^2+2b^2}=\dfrac{5a.4a}{3a^2+2.\left(4a\right)^2}=\dfrac{20a^2}{3a^2+32a^2}\)
\(=\dfrac{20a^2}{35a^2}=\dfrac{4}{7}\)
\(4a^2+b^2=5ab\)
\(\Rightarrow4a\left(a-b\right)-b\left(a-b\right)=0\)
\(\Rightarrow\left(a-b\right)\left(4a-b\right)=0\)
\(\Rightarrow b=4a\left(do.a\ne b\right)\)
\(\dfrac{5ab}{3a^2+2b^2}=\dfrac{20a^2}{3a^2+32a^2}=\dfrac{4}{7}\)
\(3a^2+4b^2=7ab\)
\(\Rightarrow3a^2+4b^2-7ab=0\)
\(\Rightarrow3a^2-3ab-4ab+4b^2=0\)
\(\Rightarrow3a\left(a-b\right)-4b\left(a-b\right)=0\)
\(\Rightarrow\left(a-b\right)\left(3a-4b\right)=0\)
Mà \(a\ne b\Rightarrow a-b\ne0\)
Từ đó \(3a-4b=0\Rightarrow3a=4b\Rightarrow a=\frac{4}{3}b\)
\(E=\frac{a+2b}{3a-b}=\frac{\frac{4}{3}b+2b}{3.\frac{4}{3}b-b}=\frac{10}{9}\)
9a2 + 4b2 = 13ab => (3a)2 + 2.3a.2b + (2b)2 = 25ab => (3a+2b)2 = 25ab => 3a + 2b = 5\(\sqrt{ab}\) (do 3a ; 2b > 0)
9a2 + 4b2 = 13ab => (3a)2 - 2.3a.2b + (2b)2 = ab => (3a- 2b)2 = ab => 3a - 2b = \(\sqrt{ab}\) (ví 3a > 2b > 0)
A = \(\frac{ab}{\left(3a-2b\right)\left(3a+2b\right)}=\frac{ab}{\sqrt{ab}.5\sqrt{ab}}=\frac{1}{5}\)
P=3a-2b\2a+5 + 3b-a\b-5
=2a+a-2b\2a-5 + -a+2b+b\b-5
=2a+(a-2b)\2a-5 + -(a-2b)+b
=2a+5\2a-5 + -5+b\b-5
=-(2a-5)\(2a-5) + (b-5)\(b-5)
=-1+1=0
