Certainly! Let's consider the expression we need to simplify:
[tex]\[
\frac{6 \cdot \sqrt[3]{4}}{2 \cdot \sqrt[3]{6}}
\][/tex]
Here's the step-by-step process to simplify this expression:
1. Identify the numerators and denominators:
- The numerator is [tex]\(6 \cdot \sqrt[3]{4}\)[/tex].
- The denominator is [tex]\(2 \cdot \sqrt[3]{6}\)[/tex].
2. Simplify the constants:
- The constants in the numerator and denominator are 6 and 2, respectively.
3. Divide the constants:
[tex]\[
\frac{6}{2} = 3
\][/tex]
So the fraction simplifies to:
[tex]\(
3 \cdot \frac{\sqrt[3]{4}}{\sqrt[3]{6}}
\)[/tex].
4. Combine the cube roots:
[tex]\[
3 \cdot \sqrt[3]{\frac{4}{6}}
\][/tex]
5. Simplify the fraction inside the cube root:
[tex]\[
\frac{4}{6} = \frac{2}{3}
\][/tex]
Therefore, our expression simplifies to:
[tex]\[
3 \cdot \sqrt[3]{\frac{2}{3}}
\][/tex]
This is the simplest form of the given expression. Upon further calculations, the value of this simplified expression is approximately:
[tex]\[
3 \cdot \sqrt[3]{\frac{2}{3}} \approx 2.6207413942088964
\][/tex]
Therefore, the simplified expression evaluates to approximately [tex]\(2.6207413942088964\)[/tex].
