Use the properties of logarithms to rewrite the expression. Simplify the result as much as possible. Assume all variables represent positive real numbers. \[ \log _{3} \frac{\sqrt{x} \sqrt[3]{y}}{w^{7} \sqrt{z}} \]

Step 1 ：Given the expression \(\log _{3} \frac{\sqrt{x} \sqrt[3]{y}}{w^{7} \sqrt{z}}\)

Step 2 ：We can use the properties of logarithms to simplify this expression.

Step 3 ：The properties of logarithms that we will use are: \(\log_b{a^n} = n \log_b{a}\), \(\log_b{\frac{a}{c}} = \log_b{a} - \log_b{c}\), and \(\log_b{a \cdot c} = \log_b{a} + \log_b{c}\)

Step 4 ：Applying these properties to the given expression, we get \(-7\log(w)/\log(3) + 0.5\log(x)/\log(3) + 0.333333333333333\log(y)/\log(3) - 0.5\log(z)/\log(3)\)

Step 5 ：Thus, the simplified expression is \(\boxed{\frac{1}{2} \log _{3} x+\frac{1}{3} \log _{3} y-7 \log _{3} w-\frac{1}{2} \log _{3} z}\)