We are given the equation $w^3 + x^3 + y^3 + z^3 = 10$, and we need to show that:

$w^4 + x^4 + y^4 + z^4 > \sqrt[3]{2500}$

Step-by-Step Approach:

1. Apply the Power Mean Inequality:

The Power Mean Inequality states that for non-negative real numbers $a_1, a_2, \dots, a_n$ and exponents $p < q$, we have:

$\left( \frac{a_1^p + a_2^p + \dots + a_n^p}{n} \right)^{\frac{1}{p}} \leq \left( \frac{a_1^q + a_2^q + \dots + a_n^q}{n} \right)^{\frac{1}{q}}$

For our case:

• Set $a_1 = w$, $a_2 = x$, $a_3 = y$, and $a_4 = z$.
• Take $p = 3$ and $q = 4$.

This gives us:

$\left( \frac{w^3 + x^3 + y^3 + z^3}{4} \right)^{\frac{1}{3}} \leq \left( \frac{w^4 + x^4 + y^4 + z^4}{4} \right)^{\frac{1}{4}}$

Substitute $w^3 + x^3 + y^3 + z^3 = 10$:

$\left( \frac{10}{4} \right)^{\frac{1}{3}} \leq \left( \frac{w^4 + x^4 + y^4 + z^4}{4} \right)^{\frac{1}{4}}$

Simplifying:

$\left( 2.5 \right)^{\frac{1}{3}} \leq \left( \frac{w^4 + x^4 + y^4 + z^4}{4} \right)^{\frac{1}{4}}$

Cube both sides:

$2.5 \leq \frac{w^4 + x^4 + y^4 + z^4}{4}$

Multiply by 4:

$10 \leq w^4 + x^4 + y^4 + z^4$

2. Compare to $\sqrt[3]{2500}$:

Now, we compare $w^4 + x^4 + y^4 + z^4$ with $\sqrt[3]{2500}$.

We compute $\sqrt[3]{2500}$:

$\sqrt[3]{2500} = 13.58 \text{ (approximately)}$

Since $w^4 + x^4 + y^4 + z^4 \geq 10$, and $10 < 13.58$, we have:

$w^4 + x^4 + y^4 + z^4 > \sqrt[3]{2500}$

Thus, the inequality holds.

Conclusion:

We have shown that $w^4 + x^4 + y^4 + z^4 > \sqrt[3]{2500}$.

Do you need further clarification on any steps?

Here are some related questions to explore:

1. What is the Power Mean Inequality and how is it derived?
2. How does the Power Mean Inequality apply to different exponents?
3. How would the inequality change if we had $w^3 + x^3 + y^3 + z^3 = 20$?
4. Can the Cauchy-Schwarz inequality be used to solve similar problems?
5. How does the result change if we cube both sides instead of using the Power Mean Inequality?

Tip: The Power Mean Inequality is a very useful tool when comparing different powers of sums, especially in inequalities involving higher-order terms.