MCQ
If $a=2^{-2}-2^{-3}, b=2^{-3}-2^{-4}$ and $c=2^{-4}-2^{-2}$, then $a^3+b^3+c^3=$
- A$-\frac{9}{1024}$
- ✓$-\frac{9}{2048}$
- C$0$
- D1
✓
Answer
Correct option: B.
$-\frac{9}{2048}$
(b)
We find that a + b + c = 0
$\begin{aligned} \therefore \quad a^3+b^3+c^3 & =3 a b c \\ & =3\left(2^{-2}-2^{-3}\right)\left(2^{-3}-2^{-4}\right)\left(2^{-4}-2^{-2}\right) \\ & =3 \times 2^{-3}(2-1) 2^{-4}(2-1) 2^{-4}\left(1-2^2\right)=3 \times 2^{-11} \times-3=-\frac{9}{2^{11}}=-\frac{9}{2048}\end{aligned}$
We find that a + b + c = 0
$\begin{aligned} \therefore \quad a^3+b^3+c^3 & =3 a b c \\ & =3\left(2^{-2}-2^{-3}\right)\left(2^{-3}-2^{-4}\right)\left(2^{-4}-2^{-2}\right) \\ & =3 \times 2^{-3}(2-1) 2^{-4}(2-1) 2^{-4}\left(1-2^2\right)=3 \times 2^{-11} \times-3=-\frac{9}{2^{11}}=-\frac{9}{2048}\end{aligned}$
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