If x is a positive real number such that x^2 + 1/x^2 = 98, then what is the value of x^3 + 1/x^3?

Introduction / Context:
This question is an algebraic manipulation problem involving symmetric expressions in x and 1/x. You are given x^2 + 1/x^2 and asked to find x^3 + 1/x^3. Problems of this type are common in aptitude tests because they test your ability to use identities and to work with powers of a variable and its reciprocal without explicitly solving for the variable.

Given Data / Assumptions:

• x is a positive real number.
• x^2 + 1/x^2 = 98.
• We need x^3 + 1/x^3.

Concept / Approach:
The standard approach is to first find x + 1/x from the given x^2 + 1/x^2 using the identity (x + 1/x)^2 = x^2 + 1/x^2 + 2. Once x + 1/x is found, we can use the identity x^3 + 1/x^3 = (x + 1/x)^3 − 3(x + 1/x). The condition x positive ensures the correct sign for x + 1/x.

Step-by-Step Solution:
We know x^2 + 1/x^2 = 98. Use (x + 1/x)^2 = x^2 + 1/x^2 + 2. So (x + 1/x)^2 = 98 + 2 = 100. Therefore x + 1/x = ±10. Given x is positive, x + 1/x is also positive, so x + 1/x = 10. Now use the identity x^3 + 1/x^3 = (x + 1/x)^3 − 3(x + 1/x). Compute (x + 1/x)^3 = 10^3 = 1000. Compute 3(x + 1/x) = 3·10 = 30. Thus x^3 + 1/x^3 = 1000 − 30 = 970.
Verification / Alternative check:
You can verify consistency by working backwards. Suppose x^3 + 1/x^3 = 970 and x + 1/x = 10. Then expanding (x + 1/x)^3 gives x^3 + 1/x^3 + 3(x + 1/x). Substituting 970 and 10 yields 970 + 30 = 1000, which matches 10^3, confirming the identities are used correctly.

Why Other Options Are Wrong:
Values such as 1030 or 980 arise from incorrectly adding terms or misapplying the cubic identity, for example using 2(x + 1/x) instead of 3(x + 1/x). The negative options would correspond to x + 1/x = −10, which contradicts the condition that x is positive, because a positive x makes x + 1/x positive.

Common Pitfalls:
One common error is to try solving the quadratic in x^2 straight away, which is unnecessary and time-consuming. Another is forgetting to add the extra 2 when going from x^2 + 1/x^2 to (x + 1/x)^2. Always recall and carefully apply the standard identities to avoid these mistakes.

Final Answer:
The value of x^3 + 1/x^3 is 970.