Question

Evaluate \( \left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6 \)

Hint:

When expanding expressions like \( (a + b)^n - (a - b)^n \), use symmetry to simplify the calculation by focusing only on the even or odd powers.

Answer 1

Step 1: Use the binomial expansion for both terms.
We will expand both \( \left( \sqrt{3} + \sqrt{2} \right)^6 \) and \( \left( \sqrt{3} - \sqrt{2} \right)^6 \) using the binomial theorem. The binomial expansion for \( (a + b)^n \) is: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] For \( \left( \sqrt{3} + \sqrt{2} \right)^6 \), the expansion is: \[ \left( \sqrt{3} + \sqrt{2} \right)^6 = \sum_{k=0}^{6} \binom{6}{k} (\sqrt{3})^{6-k} (\sqrt{2})^k \] For \( \left( \sqrt{3} - \sqrt{2} \right)^6 \), the expansion is: \[ \left( \sqrt{3} - \sqrt{2} \right)^6 = \sum_{k=0}^{6} \binom{6}{k} (\sqrt{3})^{6-k} (-\sqrt{2})^k \]
Step 2: Subtract the expansions.
When subtracting the two expansions, we observe that the odd powers of \( \sqrt{2} \) will cancel out, leaving only even powers. This simplifies to: \[ \left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6 = 2 \sum_{k \text{ even}} \binom{6}{k} (\sqrt{3})^{6-k} (\sqrt{2})^k \]
Step 3: Conclusion.
By calculating the above sum, we can find the exact value of the expression.