prove that for real a,b (a^2+b^2)*(a^4+b^4)>a^3+b^3

To prove that \((a^2 + b^2)(a^4 + b^4) > a^3 + b^3\) for real numbers \(a\) and \(b\), let's analyze both sides of the inequality.

### Step 1: Expanding the Left Side

The left side is \((a^2 + b^2)(a^4 + b^4)\), so let's expand it:
\[
(a^2 + b^2)(a^4 + b^4) = a^6 + a^2b^4 + b^2a^4 + b^6.
\]

So, the left side becomes:
\[
a^6 + b^6 + a^2b^4 + b^2a^4.
\]

### Step 2: Examining Each Term

Each term in this expression is non-negative for real \(a\) and \(b\). Also, \(a^6\) and \(b^6\) are typically larger than \(a^3\) and \(b^3\), respectively, especially as \(a\) and \(b\) grow in magnitude.

### Step 3: Comparing Terms

Now let's consider the expression on the right side, \(a^3 + b^3\), and compare it with each term in the expanded left side.

Notice:
1. \(a^6 + b^6 \geq a^3 + b^3\), because \(a^6 \geq a^3\) and \(b^6 \geq b^3\) for real \(a\) and \(b\).
2. The additional terms \(a^2b^4\) and \(b^2a^4\) on the left side are both non-negative, which makes the left side strictly larger than the right side when both \(a\) and \(b\) are nonzero.

### Conclusion

Thus, we conclude that:
\[
(a^2 + b^2)(a^4 + b^4) > a^3 + b^3
\]
for real values of \(a\) and \(b\).

Would you like a deeper breakdown of any part of this proof or have any questions?

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### Related Questions
1. How would this inequality change if we only considered positive values of \(a\) and \(b\)?
2. What is the relationship between \(a^6\) and \(a^3\) for values of \(a\) in the interval \(0 < a < 1\)?
3. How would this inequality behave if \(a\) and \(b\) were complex numbers?
4. Could we generalize this inequality to higher powers of \(a\) and \(b\)?
5. What role does symmetry in \(a\) and \(b\) play in this proof?

#### Tip
When analyzing inequalities, expanding terms and comparing each part separately can clarify which side is larger, especially when one side has additional non-negative terms.

Prove that for real a, b (a^2 + b^2)(a^4 + b^4) > a^3 + b^3.

This solution provides a step-by-step proof for the inequality (a^2 + b^2)(a^4 + b^4) > a^3 + b^3 with a focus on expanding terms and comparing sides. Key concepts include inequalities, expansion, and properties of real numbers. Suitable for advanced high school students in Grades 11-12.

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