The expression provided in the image is:

[ \sqrt[3]{-54} \cdot \sqrt[3]{\frac{1}{2}} ]

Let's break it down step by step:

1. Cube root of -54: [ \sqrt[3]{-54} = -\sqrt[3]{54} ] The cube root of 54 is approximately $\sqrt[3]{54} \approx 3.78$, so: $\sqrt[3]{-54} \approx -3.78$

2. Cube root of $\frac{1}{2}$: [ \sqrt[3]{\frac{1}{2}} = \frac{\sqrt[3]{1}}{\sqrt[3]{2}} = \frac{1}{\sqrt[3]{2}} \approx \frac{1}{1.26} \approx 0.7937 ]

3. Multiplying both values: $-3.78 \times 0.7937 \approx -2.999 \approx -3$

Thus, the value of the expression is approximately:

$\boxed{-3}$

Would you like more details or have any questions?

Here are 5 follow-up questions based on this problem:

1. How do you compute the cube root of a negative number?
2. What is the cube root of a fraction and how is it calculated?
3. Can the cube root of a number ever result in an imaginary number?
4. Why does multiplying a negative number by a positive number result in a negative number?
5. How is the cube root different from the square root in terms of properties?

Tip: When dealing with cube roots, remember that cube roots of negative numbers are also negative, unlike square roots, which are imaginary when the number is negative.