If $y$ is inversely proportional to the 4th power of $x$, it means the relationship between $y$ and $x$ can be expressed as:
$y = \frac{k}{x^4}$ where $k$ is the constant of proportionality.

Step 1: Find the constant $k$

Given: $y = 18$ when $x = 7$.

$18 = \frac{k}{7^4}$

Since $7^4 = 2401$, the equation becomes:

$18 = \frac{k}{2401} \implies k = 18 \times 2401 = 43218$

Step 2: Solve for $y$ when $x = 4$

Now, we use the equation $y = \frac{k}{x^4}$ with $k = 43218$ and $x = 4$.

$y = \frac{43218}{4^4}$

Since $4^4 = 256$, the equation becomes:

$y = \frac{43218}{256} = 168.84$

Final Answer:

When $x = 4$, the value of $y$ is:

$y = 168.84$

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Would you like further clarification or steps on this solution?

Here are 5 related questions to explore further:

1. What does "inversely proportional" mean, and how does it differ from direct proportionality?
2. How does changing the power (like from $x^2$ to $x^4$) affect the relationship?
3. What happens to $y$ if $x$ doubles or halves in such a relationship?
4. How can we visualize inverse proportionality on a graph?
5. How do we find the constant of proportionality $k$ in direct proportionality?

Tip: In inverse relationships, as the independent variable increases, the dependent variable decreases—pay close attention to the power involved!