If a is positive and a^2 + 1/a^2 = 7, then what is the value of a^3 + 1/a^3 ?

Introduction / Context: This question tests your understanding of algebraic identities involving expressions of the form a + 1/a, a^2 + 1/a^2, and a^3 + 1/a^3. Such identities are very popular in competitive exams because they allow you to avoid solving for a directly and instead use smart manipulation of formulas.

Given Data / Assumptions:

• a is a positive real number.
• We are given that a^2 + 1/a^2 = 7.
• We must find the value of a^3 + 1/a^3.

Concept / Approach: The standard approach is to introduce a new variable t = a + 1/a. Then we use algebraic identities to relate t, a^2 + 1/a^2, and a^3 + 1/a^3. The key identities are: a^2 + 1/a^2 = t^2 - 2 a^3 + 1/a^3 = t^3 - 3t Once we find t from the first identity, we substitute in the second identity to get the required value.

Step-by-Step Solution: Step 1: Let t = a + 1/a. Step 2: Use the identity a^2 + 1/a^2 = t^2 - 2. Step 3: We are given a^2 + 1/a^2 = 7, so t^2 - 2 = 7. Step 4: Solve for t^2: t^2 = 7 + 2 = 9. Step 5: Since a is positive, a + 1/a is also positive, therefore t = 3 (and not -3). Step 6: Now use the identity a^3 + 1/a^3 = t^3 - 3t. Step 7: Substitute t = 3 to get a^3 + 1/a^3 = 3^3 - 3*3. Step 8: Compute 3^3 = 27 and 3*3 = 9, so a^3 + 1/a^3 = 27 - 9 = 18.

Verification / Alternative check: We can cross check by finding a explicitly. From t = a + 1/a = 3, we have a^2 - 3a + 1 = 0. Solving this quadratic gives two positive roots, but both will yield the same value of a^2 + 1/a^2 and a^3 + 1/a^3. Substituting either root into a^3 + 1/a^3 confirms that the expression is indeed 18, which matches our identity based method.

Why Other Options Are Wrong: Option 9 corresponds to t^2, not a^3 + 1/a^3. Option 27 is t^3, which ignores the subtraction of 3t in the identity. Option 0 would suggest a^3 + 1/a^3 cancels out completely, which is not supported by the given condition. Option 7 simply repeats the value of a^2 + 1/a^2 and does not apply the correct identity to move from squares to cubes.

Common Pitfalls: A frequent error is to assume t = ±3 without using the information that a is positive. Another mistake is to try to solve directly for a^3 and 1/a^3 separately, which is unnecessary and time consuming. Some students also confuse the identities and write a^3 + 1/a^3 = t^3 - 2t, which is incorrect. Remember the correct forms and apply them step by step.

Final Answer: Therefore, when a is positive and a^2 + 1/a^2 = 7, the value of a^3 + 1/a^3 is 18.