9514 1404 393
Answer:
2
Step-by-step explanation:
A calculator can give you the answer quickly: 2.
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Working this out in detail is a little more complicated. Eliminating roots in the denominator means multiplying numerator and denominator by the "conjugate" of the denominator.
[tex]\dfrac{6}{\sqrt{36}+\sqrt{27}}+\dfrac{6}{\sqrt{27}+\sqrt{18}}+\dfrac{6}{\sqrt{18}+\sqrt{9}}\\\\=6\left(\dfrac{6-3\sqrt{3}}{36-27}+\dfrac{3(\sqrt{3}-\sqrt{2})}{27-18}+\dfrac{3(\sqrt{2}-1)}{18-9}\right)\\\\=\dfrac{6}{9}(6-3\sqrt{3}+3\sqrt{3}-3\sqrt{2}+3\sqrt{2}-3)=\dfrac{2}{3}\cdot 3=\boxed{2}[/tex]
