Simplify the expre \( \left(-105^{6}\right)^{\frac{3}{4}}= \)

To simplify the expression \(\left(-105^{6}\right)^{\frac{3}{4}}\), we can start by taking the exponent outside of the parentheses: \[ \left(-105^{6}\right)^{\frac{3}{4}} = -\left(105^{6}\right)^{\frac{3}{4}}. \] Next, we apply the power of a power rule: \[ \left(105^{6}\right)^{\frac{3}{4}} = 105^{6 \cdot \frac{3}{4}} = 105^{\frac{18}{4}} = 105^{\frac{9}{2}}. \] Now we can rewrite \(105^{\frac{9}{2}}\) in terms of a radical: \[ 105^{\frac{9}{2}} = \sqrt{105^9} = \sqrt{(105^8)(105)} = 105^4 \cdot \sqrt{105}. \] Calculating \(105^4\): \[ 105^4 = (105^2)^2 = (11025)^2 = 11025 \times 11025 = 121552625. \] Now substituting back: \[ \left(-105^{6}\right)^{\frac{3}{4}} = -105^4 \cdot \sqrt{105} = -121552625 \cdot \sqrt{105}. \] Thus the simplified expression is: \[ -\left(105^4 \cdot \sqrt{105}\right) = -121552625\sqrt{105}. \]