To simplify the expression \( \frac{3 \sqrt{8}}{\sqrt{2}} \) using properties of exponents and radicals, we can start by rewriting the square roots as exponents.
We know that: \[ \sqrt{a} = a^{1/2} \] Thus, we can rewrite the expression as: \[ \frac{3 \sqrt{8}}{\sqrt{2}} = \frac{3 \cdot 8^{1/2}}{2^{1/2}} \]
Using the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \), we can rewrite the expression: \[ \frac{8^{1/2}}{2^{1/2}} = ( \frac{8}{2} )^{1/2} \]
Now, simplify \( \frac{8}{2} = 4 \): \[ \frac{3 \sqrt{8}}{\sqrt{2}} = 3 \cdot (4^{1/2}) = 3 \cdot 2 \]
Thus, we have: \[ 3 \cdot 2 = 6 \]
So the equivalent expression for \( \frac{3 \sqrt{8}}{\sqrt{2}} \) is: \[ 6 \]
Therefore, the final answer is \( \boxed{6} \).