Fully simplify \( \left(36 n^{10} p^{5}\right)^{\frac{3}{2}} \)

Let's simplify the expression \( \left(36 n^{10} p^{5}\right)^{\frac{3}{2}} \). First, we can use the property of exponents that states \( (a \cdot b)^{m} = a^{m} \cdot b^{m} \). This allows us to distribute the exponent to each factor inside the parentheses: \[ \left(36 n^{10} p^{5}\right)^{\frac{3}{2}} = 36^{\frac{3}{2}} \cdot \left(n^{10}\right)^{\frac{3}{2}} \cdot \left(p^{5}\right)^{\frac{3}{2}}. \] Now, we will evaluate each part. 1. **Calculating \( 36^{\frac{3}{2}} \)**: \[ 36^{\frac{3}{2}} = (36^{\frac{1}{2}})^{3} = 6^{3} = 216. \] 2. **Calculating \( \left(n^{10}\right)^{\frac{3}{2}} \)**: \[ \left(n^{10}\right)^{\frac{3}{2}} = n^{10 \cdot \frac{3}{2}} = n^{15}. \] 3. **Calculating \( \left(p^{5}\right)^{\frac{3}{2}} \)**: \[ \left(p^{5}\right)^{\frac{3}{2}} = p^{5 \cdot \frac{3}{2}} = p^{\frac{15}{2}}. \] Putting all these results together, we have: \[ \left(36 n^{10} p^{5}\right)^{\frac{3}{2}} = 216 n^{15} p^{\frac{15}{2}}. \] Thus, the fully simplified expression is: \[ \boxed{216 n^{15} p^{\frac{15}{2}}}. \]