Simplify the expression $\sqrt[4]{32} - 3\sqrt[4]{2}$.

1. Problem Description

Simplify the expression $\sqrt[4]{32} - 3\sqrt[4]{2}$.

2. Solution Steps

First, we can rewrite $32$ as $2^5$. Then the expression becomes:

$\sqrt[4]{2^5} - 3\sqrt[4]{2}$

We can rewrite the first term as:

$\sqrt[4]{2^4 \cdot 2} - 3\sqrt[4]{2}$

Using the property $\sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}$, we can simplify the first term:

$\sqrt[4]{2^4} \cdot \sqrt[4]{2} - 3\sqrt[4]{2}$

Since $\sqrt[4]{2^4} = 2$, the expression becomes:

$2\sqrt[4]{2} - 3\sqrt[4]{2}$

Now, we can combine the terms since they have the same radical part:

$(2-3)\sqrt[4]{2}$

$-1\sqrt[4]{2}$

$-\sqrt[4]{2}$

3. Final Answer

$-\sqrt[4]{2}$