Question

What is the value of $a^3 + b^3$?
I. $a^2 + b^2 = 22$
II. $ab = 3$

• A. The question can be answered by one of the statements alone but not by the other.
• B. The question can be answered by using either statement alone.
• C. The question can be answered by using both the statements together, but cannot be answered by using either statement alone.
• D. The question cannot be answered even by using both statements together.

Hint:

Use the sum of cubes factorization and symmetric expressions to combine given data.

Answer 1

The Correct Option is C

We know: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$. From I: $a^2 + b^2 = 22$.
From II: $ab = 3$. Then $a^2 - ab + b^2 = 22 - 3 = 19$.
Also, $(a+b)^2 = a^2 + b^2 + 2ab = 22 + 6 = 28 \Rightarrow a+b = 2\sqrt{7}$. Thus, $a^3 + b^3 = (2\sqrt{7}) \times 19 = 38\sqrt{7}$. Both statements are required together.