Suppose that y is inversely proportional to the 4th power of x, and that y=18
 when x=7
. What is y when x=4
? Round your answer to two decimal places if necessary.

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!

Suppose that y is inversely proportional to the 4th power of x, and that y=18 when x=7. What is y when x=4? Round your answer to two decimal places if necessary.

Learn how to solve an inverse proportionality problem where y is inversely proportional to the 4th power of x. In this example, y = 18 when x = 7, and the goal is to find y when x = 4. Includes step-by-step explanation using the formula y = k / x^4.

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