If w^3+x^3+y^3+z^3=10 show that w^4+ x^4+y^4+z^4>^3✔️2500

Let's analyze the equation step by step:

We are given that:

\[
w^3 + x^3 + y^3 + z^3 = 10
\]

You are asked to show that:

\[
w^4 + x^4 + y^4 + z^4 > 2500
\]

### Step 1: Apply inequalities to analyze powers
To relate \( w^4 + x^4 + y^4 + z^4 \) to \( w^3 + x^3 + y^3 + z^3 \), we can first use basic inequalities and properties of power functions.

- Each of \( w^4, x^4, y^4, z^4 \) grows faster than \( w^3, x^3, y^3, z^3 \) respectively for values greater than 1.
- Let's explore extreme cases and possible values for \( w, x, y, z \).

### Step 2: Test critical values
If \( w = x = y = z \), the equation simplifies, and we can investigate this situation. Let’s assume:

\[
w = x = y = z = t
\]

Thus, the equation becomes:

\[
4t^3 = 10 \quad \Rightarrow \quad t^3 = \frac{10}{4} = 2.5 \quad \Rightarrow \quad t = \sqrt[3]{2.5}
\]

Now, we compute the value of \( w^4 + x^4 + y^4 + z^4 \):

\[
4t^4 = 4 \times \left( \sqrt[3]{2.5} \right)^4 = 4 \times \left( 2.5^{4/3} \right)
\]

Let’s estimate \( 2.5^{4/3} \).

### Step 3: Approximate calculation

\[
2.5^{4/3} \approx 3.9528
\]

Therefore:

\[
4t^4 = 4 \times 3.9528 \approx 15.8112
\]

This does not satisfy the inequality \( w^4 + x^4 + y^4 + z^4 > 2500 \). However, it seems the problem might involve further investigation or clarification, particularly with the inequality symbol or context provided.

Could you please clarify the inequality or confirm if there was a typographical error?

If w^3 + x^3 + y^3 + z^3 = 10, show that w^4 + x^4 + y^4 + z^4 > 2500.

Solve this advanced inequality problem where w^3 + x^3 + y^3 + z^3 = 10, and show that w^4 + x^4 + y^4 + z^4 > 2500. This problem explores algebraic concepts, inequalities, and power functions, suitable for Grades 10-12.

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