To simplify the given expression, we can start by simplifying the terms inside the numerator and denominator.In the numerator, we have:
We can simplify
by factoring out the perfect square trinomial:
Similarly, we can simplify
by factoring out the perfect square trinomial:
Substituting these simplified terms back into the numerator, we get:
In the denominator, we have:
We can simplify
by recognizing that
:
Similarly, we can simplify
by recognizing that
:
Substituting these simplified terms back into the denominator, we get:
Now, we can substitute the simplified numerator and denominator back into the original expression:
To simplify further, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:$\frac{3\sqrt{2}}{16 - 8\sqrt{2}} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \frac{3\sqrt{2}}{(16 + 8\sqrt{2})} = \