To express the fraction [tex]\(\frac{\sqrt[3]{32}}{5 \sqrt[3]{4}}\)[/tex] as a pure surd, let's take it step-by-step:
1. Simplify the Fraction:
We start with the expression:
[tex]\[
\frac{\sqrt[3]{32}}{5 \sqrt[3]{4}}
\][/tex]
2. Calculate the Cube Roots:
- The cube root of 32 is [tex]\(\sqrt[3]{32}\)[/tex].
- The cube root of 4 is [tex]\(\sqrt[3]{4}\)[/tex].
From the previous result, we have:
[tex]\(\sqrt[3]{32} \approx 3.1748\)[/tex]
[tex]\(\sqrt[3]{4} \approx 1.5874\)[/tex]
3. Simplify the Components:
- For the numerator [tex]\(\sqrt[3]{32}\)[/tex], we get approximately 3.1748.
- For the denominator [tex]\(5 \sqrt[3]{4}\)[/tex], since [tex]\(\sqrt[3]{4}\)[/tex] is approximately 1.5874, we multiply:
[tex]\[
5 \cdot 1.5874 \approx 7.937
\][/tex]
4. Form the Fraction:
Now, divide these simplified components:
[tex]\[
\frac{3.1748}{7.937} = 0.4
\][/tex]
5. Convert the Fraction to a Pure Surd:
We need to express [tex]\(0.4\)[/tex] as a cube root. Notice:
[tex]\[
0.4 = \frac{\sqrt[3]{32}}{5 \sqrt[3]{4}} = \sqrt[3]{\frac{32}{125}}
\][/tex]
Since [tex]\( \frac{32}{125} = \frac{8}{25} \)[/tex], we get:
[tex]\[
\frac{\sqrt[3]{32}}{5 \sqrt[3]{4}} = \sqrt[3]{\frac{8}{25}}
\][/tex]
Conclusion: The correct expression of [tex]\(\frac{\sqrt[3]{32}}{5 \sqrt[3]{4}}\)[/tex] as a pure surd is:
[tex]\[
\sqrt[3]{\frac{8}{25}}
\][/tex]
Therefore, the correct answer is option 2:
[tex]\(\sqrt[3]{\frac{8}{25}}\)[/tex].
