Fully simplify \( \left(36 n^{10} p^{6}\right)^{\frac{1}{2}} \)

To simplify \( \left(36 n^{10} p^{6}\right)^{\frac{1}{2}} \), we start by applying the power of a product property, which allows us to take the square root of each factor separately: \[ \left(36 n^{10} p^{6}\right)^{\frac{1}{2}} = \left(36\right)^{\frac{1}{2}} \cdot \left(n^{10}\right)^{\frac{1}{2}} \cdot \left(p^{6}\right)^{\frac{1}{2}} \] Calculating each part: 1. \(\left(36\right)^{\frac{1}{2}} = 6\) 2. \(\left(n^{10}\right)^{\frac{1}{2}} = n^{10 \cdot \frac{1}{2}} = n^{5}\) 3. \(\left(p^{6}\right)^{\frac{1}{2}} = p^{6 \cdot \frac{1}{2}} = p^{3}\) Putting it all together, we get: \[ \left(36 n^{10} p^{6}\right)^{\frac{1}{2}} = 6 n^{5} p^{3} \] Thus, the fully simplified expression is: \[ \boxed{6 n^{5} p^{3}} \]