Question

If a + b = 4 and $\frac{1}{a}+\frac{1}{b}=4$, then find the value of $a^3+b^3$.

• A. $4$
• B. $64$
• C. $52$
• D. $25$
• E. None of these

Answer 1

The correct option is C $52$

Given :

$a+b=4$...........[1]

$\frac{1}{a}+\frac{1}{b}=4⟹a+b=4ab$...........[2].

From $1$ and $2$:

$4=4ab⟹ab=1$

Now squaring equation $1$, we get

$a^2+b^2+2ab=16$

$⟹a^2+b^2=14$.......[3]

Now,

$a^3+b^3=(a+b)(a^2+b^2-ab)=\left(4\right)(14-1)=4\times 13=52$