8. The geometric mean of a set of $n$ positive numbers is defined as the $n$-th root of the product of those numbers.
Yasmeen writes down a set of four numbers which have a geometric mean of 2048 .
Zak writes down a set of four numbers which have a geometric mean of 8 .
What is the geometric mean of the combined set of the eight numbers written by Yasmeen and Zak?
SOLUTION
128

Let Yasmeen's numbers be $y_{1}, y_{2}, y_{3}$ and $y_{4}$; and Zak's be $z_{1}, z_{2}, z_{3}$ and $z_{4}$.
We are given that $\sqrt[4]{y_{1} y_{2} y_{3} y_{4}}=2048$ and $\sqrt[4]{z_{1} z_{2} z_{3} z_{4}}=8$.
Therefore $\sqrt[4]{y_{1} y_{2} y_{3} y_{4} z_{1} z_{2} z_{3} z_{4}}=\sqrt[4]{y_{1} y_{2} y_{3} y_{4}} \times \sqrt[4]{z_{1} z_{2} z_{3} z_{4}}=2048 \times 8=2^{11} \times 2^{3}=2^{14}$.
We now take the square-root of both sides getting $\sqrt[8]{x_{1} x_{2} x_{3} x_{4} y_{1} y_{2} y_{3} y_{4}}=2^{7}=128$.