Find the value of (1 + √2)⁵ + (1 - √2)⁵.

Step 1. The two expansions. The terms of $(1 + \sqrt{2})^5$ are ${}^5C_r(\sqrt{2})^r$. In $(1 - \sqrt{2})^5$ each term with odd $r$ changes sign.

Step 2. Add. The odd terms cancel, and the even terms double: $2\left[{}^5C_0 + {}^5C_2(\sqrt{2})^2 + {}^5C_4(\sqrt{2})^4\right]$.

Step 3. Work it out. $2[1 + 10 \times 2 + 5 \times 4] = 2 \times 41 = 82$.