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

To fully simplify \( \left(36 n^{10} p^{5}\right)^{\frac{3}{2}} \), follow these steps: 1. **Apply the Exponent to Each Factor:** \[ \left(36 n^{10} p^{5}\right)^{\frac{3}{2}} = 36^{\frac{3}{2}} \cdot \left(n^{10}\right)^{\frac{3}{2}} \cdot \left(p^{5}\right)^{\frac{3}{2}} \] 2. **Simplify Each Term Individually:** - **For \( 36^{\frac{3}{2}} \):** \[ 36 = 6^2 \quad \text{so} \quad 36^{\frac{3}{2}} = (6^2)^{\frac{3}{2}} = 6^{3} = 216 \] - **For \( \left(n^{10}\right)^{\frac{3}{2}} \):** \[ \left(n^{10}\right)^{\frac{3}{2}} = n^{10 \cdot \frac{3}{2}} = n^{15} \] - **For \( \left(p^{5}\right)^{\frac{3}{2}} \):** \[ \left(p^{5}\right)^{\frac{3}{2}} = p^{5 \cdot \frac{3}{2}} = p^{\frac{15}{2}} = p^{7} \cdot p^{\frac{1}{2}} = p^{7} \sqrt{p} \] 3. **Combine All Simplified Terms:** \[ 216 \cdot n^{15} \cdot p^{7} \sqrt{p} \] **Final Simplified Form:** \[ 216\, n^{15}\, p^{7}\, \sqrt{p} \] **Answer:** After simplifying, the expression is 216 n¹⁵ p⁷ multiplied by √p. Thus, 216 n^15 p^7 √p