If $(a + \frac{1}{a}) = 6$, then $(a^4 + \frac{1}{a^4}) = x$

## Concept & Formula This problem tests the concept of successive squaring of algebraic identities. When given $x + \frac{1}{x} = k$, squaring both sides yields the foundational formula: $$x^2 + \frac{1}{x^2} = k^2 - 2$$ ## Step-by-Step Solution * **Given:** $(a + \frac{1}{a}) = 6$ * **Step 1: Find the square.** Apply the formula to find $(a^2 + \frac{1}{a^2})$. $a^2 + \frac{1}{a^2} = 6^2 - 2$ $a^2 + \frac{1}{a^2} = 36 - 2 = 34$ * **Step 2: Square again to reach the fourth power.** Treat $(a^2 + \frac{1}{a^2}) = 34$ as your new base equation and apply the exact same formula. $a^4 + \frac{1}{a^4} = (34)^2 - 2$ * **Step 3: Calculate the final value.** $34^2 = 1156$ $1156 - 2 = 1154$ ## Exam Strategy & Shortcut Use the unit digit trick to save calculation time. You know the sequence is $k^2 - 2$. First step: $6^2 - 2 = 34$. (Unit digit is 4). Second step: $(...4)^2 - 2 = 16 - 2 = 14$. The final answer must have a unit digit of **4**. Looking at the options, both (a) 1154 and (d) 1164 end in 4. However, $30^2$ is 900, so $34^2$ is clearly in the 1100 range, easily pointing to 1154. ## Common Pitfall Students often mistakenly square the result of the first step but forget to subtract the $2$ at the very end, selecting 1156 if it were an option. Always remember that the $+ 2(a^2)(\frac{1}{a^2})$ term needs to be moved to the other side of the equation. ## Final Answer **Therefore, the correct answer is 1154.**