Question : The cube of the sum of two given numbers is 1728, while the product of the two given numbers is 32. Find the sum of the cubes of the two given numbers.
Option 1: 576
Option 2: 250
Option 3: 640
Option 4: 512
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Correct Answer: 576
Solution :
Let the 2 numbers be $a$ and $b$.
According to the question,
$(a+b)^3 = 1728$
⇒ $a+b = 12$
Also, $a\times b = 32$
We have to find, $a^3 + b^3$
$(a+b)^3 = 1728$
⇒ $a^3 + b^3 + 3ab(a+b) = 1728$
⇒ $a^3 + b^3 = 1728 - 3ab(a+b)$
⇒ $a^3 + b^3 = 1728 - 3\times 32 \times 12$
⇒ $a^3 + b^3 = 1728-1152 = 576$
Hence, the correct answer is 576.
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