Question: ((49x^{-16}y^{-12})^{\frac{3}{2}}\)…

\((49x^{-16}y^{-12})^{\frac{3}{2}}\)

Solution

To simplify the expression \((49x^{-16}y^{-12})^{\frac{3}{2}}\), we can proceed as follows: Apply the exponent \(\frac{3}{2}\) to each part inside the parentheses: \[ 49^{\frac{3}{2}}, \quad (x^{-16})^{\frac{3}{2}}, \quad (y^{-12})^{\frac{3}{2}} \] Calculate \(49^{\frac{3}{2}}\). Since \(49 = 7^2\), we have: \[ (7^2)^{\frac{3}{2}} = 7^{2 \cdot \frac{3}{2}} = 7^3 = 343 \] Simplify \((x^{-16})^{\frac{3}{2}}\) using the power of a power property: \[ (x^{-16})^{\frac{3}{2}} = x^{-16 \cdot \frac{3}{2}} = x^{-24} \] Simplify \((y^{-12})^{\frac{3}{2}}\) using the power of a power property: \[ (y^{-12})^{\frac{3}{2}} = y^{-12 \cdot \frac{3}{2}} = y^{-18} \] Combine these results to get the simplified expression: \[ 343x^{-24}y^{-18} \]