The expression you provided contains some unclear symbols, specifically ">$^3☑✔️2500$." I assume you're asking to prove that $w^4 + x^4 + y^4 + z^4$ is greater than some number, possibly 2500, or another inequality.

Could you clarify the specific inequality or goal? For now, I'll proceed with solving the equation $w^3 + x^3 + y^3 + z^3 = 10$ and explore what we can deduce about $w^4 + x^4 + y^4 + z^4$.

Step 1: Assume some possible values for $w$, $x$, $y$, and $z$.

To satisfy $w^3 + x^3 + y^3 + z^3 = 10$, let’s test a few simple cases:

1. Case 1: $w = x = y = z$
   • Let $w = x = y = z = a$.
   • Then $4a^3 = 10$, which gives $a^3 = \frac{10}{4} = 2.5$, and hence $a = \sqrt[3]{2.5}$.
   • Now, $w^4 + x^4 + y^4 + z^4 = 4a^4$, so $a^4 = \left( \sqrt[3]{2.5} \right)^4$.

Let me calculate the value of $a^4$.For $w = x = y = z = \sqrt[3]{2.5}$, we find that $w^4 + x^4 + y^4 + z^4 = 13.57$.

This value is clearly much smaller than 2500. Therefore, if you're aiming to show that $w^4 + x^4 + y^4 + z^4$ is greater than 2500, we need to reconsider the form of your inequality. Could you confirm or clarify the specific target number or inequality?

Let me know if you'd like more details or further clarifications. Here are 5 related questions:

1. What specific inequality do you want to prove involving $w^4 + x^4 + y^4 + z^4$?
2. Should we consider specific values for $w, x, y, z$, or are we looking for a general proof?
3. Would you like to explore the relationship between higher powers of sums like this?
4. Could you clarify the role of the symbols ☑✔️ in your question?
5. Would you like to explore inequalities involving sums of cubes and fourth powers in general?

Tip: When dealing with power sums, it's useful to try simple symmetric cases (like $w = x = y = z$) to gain intuition.