Tử:
\(M=\sqrt{a+4\sqrt{a-4}}+\sqrt{a-4\sqrt{a-4}}\)
\(M^2=a+4\sqrt{a-4}+2\sqrt{\left(a+4\sqrt{a-4}\right)\left(a-4\sqrt{a-4}\right)}+a-4\sqrt{a-4}\)
\(=2a+2\sqrt{a^2-16a+64}\)
\(=2a+2\sqrt{\left(a-8\right)^2}=2a+2a-16=4a-16\)
Mẫu:
\(\sqrt{1-\dfrac{8}{a}+\dfrac{16}{a^2}}=\sqrt{\left(1-\dfrac{4}{a}\right)^2}=1-\dfrac{4}{a}\)
Ta có:
\(\dfrac{4a-16}{1-\dfrac{4}{a}}=\dfrac{4\left(a-4\right)}{\dfrac{a-4}{a}}=4a\)
\(=\dfrac{\sqrt{a+4\sqrt{a-4}}+\sqrt{a-4\sqrt{a-4}}}{\sqrt{1-\dfrac{8}{a}+\dfrac{16}{a^2}}}\)
\(=\dfrac{\sqrt{\left(\sqrt{a-4}+2\right)^2}+\sqrt{\left(\sqrt{a-4}\right)-2}}{\sqrt{\left(1-\dfrac{4}{a}\right)^2}}\)
\(=\dfrac{\sqrt{a-4}+2+\sqrt{a-4}-2}{1-\dfrac{4}{a}}\)
\(=\dfrac{2a}{\sqrt{a-4}}\)
Hok tốt!
