$\dfrac{\sqrt[]{\sqrt[]{7}-1}}{\sqrt[]{6\sqrt[]{7}+6}}$
$=\dfrac{\sqrt[]{\sqrt[]{7}-1}}{\sqrt[]{6(\sqrt[]{7}+1)}}$
$=\dfrac{\sqrt[]{\sqrt[]{7}-1}.\sqrt[]{\sqrt[]{7}-1}}{\sqrt[]{6(\sqrt[]{7}+1)}.\sqrt[]{\sqrt[]{7}-1}}$
$=\dfrac{\sqrt[]{(\sqrt[]{7}-1)^2}}{\sqrt[]{6[(\sqrt[]{7})^2-1]}}$
$=\dfrac{|\sqrt[]{7}-1|}{\sqrt[]{6.6}}$
$=\dfrac{\sqrt[]{7}-1}{\sqrt[]{6^2}}$
$=\dfrac{\sqrt[]{7}-1}{6}$
