Question : If $a + b = 6$ and $ab = 5$, then what is the value of $a^3 + b^3$?
Option 1: 106
Option 2: 136
Option 3: 126
Option 4: 116
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Correct Answer: 126
Solution :
Given: The values of $a + b = 6$ and $ab = 5$.
We know the algebraic identities, $(a+b)^2=a^2 + b^2 +2ab$ and $a^3 + b^3=(a+b)(a^2 + b^2–ab)$.
$a + b = 6$ (equation 1)
Squaring on both sides of the equation (1),
⇒ $(a + b)^2 = 6^2$
⇒ $a^2 + b^2 +2ab = 36$
⇒ $a^2 + b^2 +2\times 5 = 36$
⇒ $a^2 + b^2 = 36–10$
⇒ $a^2 + b^2 = 26$ (equation 2)
Substitute the values from equation (1) and equation (2) in the identity, $a^3 + b^3=(a+b)(a^2 + b^2–ab)$.
⇒ $a^3 + b^3=6\times(26–5)$
⇒ $a^3 + b^3=6\times21$
⇒ $a^3 + b^3=126$
Hence, the correct answer is 126.
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