Difficulty: Medium
Correct Answer: 18
Explanation:
Introduction / Context:
This question focuses on manipulating expressions involving powers of a variable and using identities for a + 1/a and higher powers such as a^3 + 1/a^3. It is a standard pattern in algebra that appears in many aptitude tests.
Given Data / Assumptions:
Concept / Approach:
First we recognize that (a^2 + 1)/a can be written as a + 1/a. From the value of a + 1/a we can compute a^2 + 1/a^2 and then a^3 + 1/a^3 using standard algebraic identities. Finally, note that (a^6 + 1)/a^3 equals a^3 + 1/a^3.
Step-by-Step Solution:
Step 1: Rewrite the given expression: (a^2 + 1)/a = a + 1/a.
Step 2: From the question, a + 1/a = 3.
Step 3: Use the identity a^2 + 1/a^2 = (a + 1/a)^2 - 2.
Step 4: Substitute a + 1/a = 3 to get a^2 + 1/a^2 = 3^2 - 2 = 9 - 2 = 7.
Step 5: Use the identity a^3 + 1/a^3 = (a + 1/a)(a^2 + 1/a^2 - 1).
Step 6: Substitute values: a^3 + 1/a^3 = 3(7 - 1) = 3 × 6 = 18.
Step 7: Observe that (a^6 + 1)/a^3 = a^3 + 1/a^3, so the required value is 18.
Verification / Alternative check:
We can solve the quadratic a + 1/a = 3 by multiplying both sides by a to obtain a^2 - 3a + 1 = 0 and solving for a. Any root of this equation, when substituted into (a^6 + 1)/a^3, produces 18. Since our algebraic simplification is independent of the specific root, the result is consistent.
Why Other Options Are Wrong:
Option A (9) might occur if a student stops at a^2 + 1/a^2. Option C (27) can appear if someone mistakenly multiplies 3 and 9 instead of using the correct identity. Option D (1) and Option E (36) are not compatible with the identities and come from algebraic slips or incorrect squaring.
Common Pitfalls:
The main pitfalls involve misremembering the identities for a^2 + 1/a^2 or a^3 + 1/a^3, or forgetting the minus sign in (a + 1/a)^2 - 2. Some learners also incorrectly think that (a^6 + 1)/a^3 equals (a^3 + 1)^2/a^3, which is not true.
Final Answer:
Therefore, the value of (a^6 + 1)/a^3 is 18.
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