Question

$\frac{2^{1001}+2^{999}}{2^{1000}-2^{998}}=?$

• A. 1
• B. 3
• C. 5
• D. ഇവയൊന്നുമല്ല

Answer 1

Answer:

D. ഇവയൊന്നുമല്ല

The value of the expression is $\frac{10}{3}$ (or $3\frac{1}{3}$).

1. Identify the smallest power of 2
To simplify the fraction easily, look for the smallest exponent in the entire expression, which is $998$. We can factor out $2^{998}$ from both the numerator and the denominator.

2. Factor the numerator
$2^{1001} + 2^{999} = 2^{998} \cdot (2^3 + 2^1)$
$= 2^{998} \cdot (8 + 2)$
$= 2^{998} \cdot 10$

3. Factor the denominator
$2^{1000} - 2^{998} = 2^{998} \cdot (2^2 - 1)$
$= 2^{998} \cdot (4 - 1)$
$= 2^{998} \cdot 3$

4. Simplify the fraction
Now substitute these back into the original problem:
$\frac{2^{1001}+2^{999}}{2^{1000}-2^{998}} = \frac{2^{998} \cdot 10}{2^{998} \cdot 3}$

Cancel out the common $2^{998}$ term from both the top and the bottom:
$= \frac{10}{3}$