To evaluate $(\sqrt{5} + \sqrt{3})^4 - (\sqrt{5} - \sqrt{3})^4$ using the binomial expansion, we proceed as follows:

Step 1: Write the Binomial Expansion

For any $(a + b)^n$, the expansion is: $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

For $(\sqrt{5} + \sqrt{3})^4$, we have: $(\sqrt{5} + \sqrt{3})^4 = \binom{4}{0} (\sqrt{5})^4 (\sqrt{3})^0 + \binom{4}{1} (\sqrt{5})^3 (\sqrt{3})^1 + \binom{4}{2} (\sqrt{5})^2 (\sqrt{3})^2 + \binom{4}{3} (\sqrt{5})^1 (\sqrt{3})^3 + \binom{4}{4} (\sqrt{5})^0 (\sqrt{3})^4$

Similarly, for $(\sqrt{5} - \sqrt{3})^4$, we use: $(\sqrt{5} - \sqrt{3})^4 = \binom{4}{0} (\sqrt{5})^4 (-\sqrt{3})^0 + \binom{4}{1} (\sqrt{5})^3 (-\sqrt{3})^1 + \binom{4}{2} (\sqrt{5})^2 (-\sqrt{3})^2 + \binom{4}{3} (\sqrt{5})^1 (-\sqrt{3})^3 + \binom{4}{4} (\sqrt{5})^0 (-\sqrt{3})^4$

Step 2: Combine Both Expansions

We subtract $(\sqrt{5} - \sqrt{3})^4$ from $(\sqrt{5} + \sqrt{3})^4$. Only the odd-power terms remain because the even-power terms cancel out. Thus: $(\sqrt{5} + \sqrt{3})^4 - (\sqrt{5} - \sqrt{3})^4 = 2 \left[ \binom{4}{1} (\sqrt{5})^3 (\sqrt{3}) + \binom{4}{3} (\sqrt{5}) (\sqrt{3})^3 \right]$

Step 3: Simplify Each Term

• $\binom{4}{1} = 4$, $\binom{4}{3} = 4$
• $(\sqrt{5})^3 = 5\sqrt{5}, (\sqrt{3})^3 = 3\sqrt{3}$

Substituting: $2 \left[ 4 \cdot 5\sqrt{5} \cdot \sqrt{3} + 4 \cdot \sqrt{5} \cdot 3\sqrt{3} \right]$ $= 2 \left[ 20\sqrt{15} + 12\sqrt{15} \right]$ $= 2 \cdot 32\sqrt{15}$ $= 64\sqrt{15}$

Final Answer:

$(\sqrt{5} + \sqrt{3})^4 - (\sqrt{5} - \sqrt{3})^4 = 64\sqrt{15}$

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Would you like me to break down any step further or verify the calculations?

Here are some related questions for exploration:

1. What is the general formula for the binomial expansion?
2. How do we determine which terms cancel when subtracting two binomial expansions?
3. Can you calculate $(\sqrt{5} + \sqrt{3})^6 - (\sqrt{5} - \sqrt{3})^6$?
4. How is symmetry in exponents used in such calculations?
5. Can this approach be extended to $(\sqrt{5} + \sqrt{3})^{2n} - (\sqrt{5} - \sqrt{3})^{2n}$?

Tip: For problems involving differences of symmetric powers, focus on odd-power terms since even-power terms cancel out.