Sure, let's solve the expression [tex]\(\sqrt[4]{32 x^{20} z^7}\)[/tex] step by step.
### Step 1: Understanding the Fourth Root
The fourth root of a number or expression can be written as the number or expression raised to the power of [tex]\(\frac{1}{4}\)[/tex]. So, we can break down [tex]\(\sqrt[4]{32 x^{20} z^7}\)[/tex] into simpler parts that each can be raised to the power of [tex]\(\frac{1}{4}\)[/tex].
### Step 2: Breaking Down the Expression
We need to take the fourth root of each component in the product separately:
[tex]\[
\sqrt[4]{32 x^{20} z^7} = \sqrt[4]{32} \cdot \sqrt[4]{x^{20}} \cdot \sqrt[4]{z^7}
\][/tex]
### Step 3: Simplifying Each Component
1. Simplifying [tex]\(\sqrt[4]{32}\)[/tex]:
The fourth root of 32 is equal to [tex]\(32^{\frac{1}{4}}\)[/tex]. Evaluating this gives us approximately [tex]\(2.378414230005442\)[/tex].
2. Simplifying [tex]\(\sqrt[4]{x^{20}}\)[/tex]:
The fourth root of [tex]\(x^{20}\)[/tex] can be written as [tex]\(x^{20 \cdot \frac{1}{4}} = x^{5}\)[/tex].
3. Simplifying [tex]\(\sqrt[4]{z^7}\)[/tex]:
The fourth root of [tex]\(z^7\)[/tex] can be written as [tex]\(z^{7 \cdot \frac{1}{4}} = z^{1.75}\)[/tex].
### Step 4: Combining the Simplified Results
Now, we combine these results back together:
[tex]\[
\sqrt[4]{32 x^{20} z^7} = 2.378414230005442 \cdot x^{5} \cdot z^{1.75}
\][/tex]
Thus, the simplified form of the original expression [tex]\(\sqrt[4]{32 x^{20} z^7}\)[/tex] is:
[tex]\[
2.378414230005442 \cdot x^{5} \cdot z^{1.75}
\][/tex]
