Ta có : \(P=\frac{a^3-a-2b-\frac{b^2}{a}}{\left(\frac{1}{\sqrt{a}}-\sqrt{\frac{1}{a}+\frac{b}{a^2}}\right)\left(\sqrt{a}+\sqrt{a+b}\right)}:\left(\frac{a^3+a^2+ab+a^2b}{a^2-b^2}+\frac{b}{a-b}\right)\)
=> \(P=\frac{\frac{a^4}{a}-\frac{a^2}{a}-\frac{2ab}{a}-\frac{b^2}{a}}{\left(\frac{1}{\sqrt{a}}-\sqrt{\frac{1}{a}+\frac{b}{a^2}}\right)\left(\sqrt{a}+\sqrt{a+b}\right)}:\left(\frac{a^2\left(a+1\right)+ab\left(a+1\right)}{\left(a-b\right)\left(a+b\right)}+\frac{b}{a-b}\right)\)
=> \(P=\frac{\frac{a^4-a^2-2ab-b^2}{a}}{\frac{\sqrt{a}}{\sqrt{a}}-\sqrt{a\left(\frac{1}{a}+\frac{b}{a^2}\right)}+\sqrt{\frac{a+b}{a}}-\sqrt{\left(a+b\right)\left(\frac{1}{a}+\frac{b}{a^2}\right)}}:\left(\frac{a\left(a+b\right)\left(a+1\right)}{\left(a-b\right)\left(a+b\right)}+\frac{b}{a-b}\right)\)
=> \(P=\frac{\frac{a^4-\left(a^2+2ab+b^2\right)}{a}}{1-\sqrt{\frac{a}{a}+\frac{ab}{a^2}}+\sqrt{\frac{a+b}{a}}-\sqrt{\frac{a}{a}+\frac{b}{a}+\frac{ab}{a^2}+\frac{b^2}{a^2}}}:\left(\frac{a\left(a+1\right)+b}{a-b}\right)\)
=> \(P=\frac{\frac{a^4-\left(a^2+2ab+b^2\right)}{a}}{1-\sqrt{1+\frac{b}{a}}+\sqrt{\frac{a+b}{a}}-\sqrt{1+\frac{2b}{a}+\frac{b^2}{a^2}}}:\left(\frac{a\left(a+1\right)+b}{a-b}\right)\)
=> \(P=\frac{\frac{a^4-\left(a+b\right)^2}{a}\left(a-b\right)}{\left(1-\sqrt{1+\frac{b}{a}}+\sqrt{\frac{a+b}{a}}-\left(\frac{b}{a}+1\right)\right)\left(a\left(a+1\right)+b\right)}\)
=> \(P=\frac{\frac{\left(a^2-a-b\right)\left(a^2+a+b\right)\left(a-b\right)}{a}}{\left(1-\frac{b}{a}-1\right)\left(a\left(a+1\right)+b\right)}\)\(=\frac{\frac{\left(a^2-a-b\right)\left(a^2+a+b\right)\left(a-b\right)}{a}}{\frac{b\left(a^2+a+b\right)}{a}}\)\(=\frac{\left(a^2-a-b\right)\left(a^2+a+b\right)\left(a-b\right)}{b\left(a^2+a+b\right)}\)
=> \(P=\frac{\left(a^2-a-b\right)\left(a-b\right)}{b}\)
- Thay a = 23, b = 22 vào biểu thức trên ta được :
\(P=\frac{\left(23^2-23-22\right)\left(23-22\right)}{22}=22\)