Simplify.
\[
\sqrt{45 x^{5} z^{8}}
\]
Assume that all variables represent positive real numbers.

Step 1 ：Break down the numbers and variables inside the square root into their prime factors.

Step 2 ：Pair up the factors and move them outside of the square root.

Step 3 ：The prime factors of 45 are 3, 3, and 5. The variable x has 5 factors and the variable z has 8 factors.

Step 4 ：Pair up the factors of 3, x, and z, and move them outside of the square root. The remaining factors that cannot be paired up will stay inside the square root, resulting in \(3\sqrt{5}x^{2}z^{4}\sqrt{x}\).

Step 5 ：Further simplify this expression by pairing up the remaining factors of x and z.

Step 6 ：For the variable x, pair up 4 of them and move them outside of the square root, leaving 1 factor of x inside the square root.

Step 7 ：For the variable z, pair up all of them and move them outside of the square root.

Step 8 ：The final simplified form of the given expression is \(\boxed{\sqrt{5} x^{5/2} z^{4}}\).