Simplify. sqrt{64 w^{6}} Assume that the variable w represents a positive real number.

To simplify \( \sqrt{64 w^{6}} \), we can break it down into two parts: the square root of 64 and the square root of \( w^{6} \). The square root of 64 is 8, since \( 8 \times 8 = 64 \). For the \( w^{6} \), we use the property of square roots that states \( \sqrt{a^m} = a^{m/2} \). Thus, \( \sqrt{w^{6}} = w^{6/2} = w^{3} \). Putting it all together, we have: \[ \sqrt{64 w^{6}} = \sqrt{64} \cdot \sqrt{w^{6}} = 8 \cdot w^{3} = 8w^{3}. \] So the final simplified form is: \[ 8w^{3}. \]