Difficulty: Medium
Correct Answer: 154/27
Explanation:
Introduction / Context:
This algebra problem shows how to move from information about x^2 + 1 / x^2 to a higher power expression x^3 + 1 / x^3. It uses standard identities that relate sums of powers of x and its reciprocal to lower degree expressions. This technique is often tested in aptitude exams to see whether you can use these identities systematically rather than trying to solve for x directly.
Given Data / Assumptions:
Concept / Approach:
The standard plan is to introduce t = x + 1 / x. There is a known relation (x + 1 / x)^2 = x^2 + 2 + 1 / x^2, which allows you to find t from the given x^2 + 1 / x^2. Once t is known, you use the identity x^3 + 1 / x^3 = (x + 1 / x)^3 − 3(x + 1 / x) to reach the target expression. This approach avoids solving a quartic equation and keeps the algebra manageable.
Step-by-Step Solution:
Step 1: Let t = x + 1 / x.Step 2: Recall that t^2 = x^2 + 2 + 1 / x^2.Step 3: You are given x^2 + 1 / x^2 = 31 / 9, so substitute into the relation: t^2 = 31 / 9 + 2.Step 4: Write 2 as 18 / 9. Then t^2 = 31 / 9 + 18 / 9 = 49 / 9.Step 5: Since x > 0, both x and 1 / x are positive, so t = x + 1 / x is positive. Therefore t = 7 / 3 (the positive square root of 49 / 9).Step 6: Use the identity x^3 + 1 / x^3 = t^3 − 3t.Step 7: Compute t^3 = (7 / 3)^3 = 343 / 27.Step 8: Compute 3t = 3 * 7 / 3 = 7.Step 9: Write 7 as a fraction with denominator 27: 7 = 189 / 27.Step 10: Therefore x^3 + 1 / x^3 = 343 / 27 − 189 / 27 = (343 − 189) / 27 = 154 / 27.Verification / Alternative check:
You can verify the consistency by working backwards. If x^3 + 1 / x^3 = 154 / 27, and t = 7 / 3, then t^3 − 3t should reproduce 154 / 27, which it does. Additionally, squaring t gives back t^2 = 49 / 9, and subtracting 2 yields x^2 + 1 / x^2 = 31 / 9, confirming that all identities are used correctly.
Why Other Options Are Wrong:
The values 70/9, 349/27, 349/7, and 70/27 correspond to partial steps or arithmetic mistakes. For example, 343 / 27 without subtracting 3t, or subtracting 3t with the wrong denominator, leads to some of these incorrect options. Only 154 / 27 carefully follows the identity and the required fraction operations.
Common Pitfalls:
Students sometimes forget to include the plus 2 in the formula t^2 = x^2 + 2 + 1 / x^2, or they take the wrong sign for t after taking the square root. Others make fraction errors when converting 7 to 189 / 27. Keeping track of denominators and signs at each step is essential for an accurate result.
Final Answer:
The value of x^3 + 1 / x^3 is 154/27.
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