Question

Refer this attachment ​

Answer 1

Question:

The value of (2^2001 + 2^1999)/(2^2000 - 2^1998) is ____.

Answer:

(B) 10/3

Solution:

$\sf \dfrac{ {2}^{2001} + {2}^{1999} }{ {2}^{2000} - {2}^{1998} }$

Take the number with lesser exponent as a common factor.

$= \sf{ \dfrac{ {2}^{1999}( {2}^{2} + 1) }{ {2}^{1998} ( {2}^{2} - 1)} }$

On simplifying,

$= \: \sf{ \dfrac{ {2}^{1999} (4 + 1)}{ {2}^{1998} (4 - 1)} }$

$\implies \sf{ \dfrac{ {2}^{1999} (5)}{ {2}^{1998} (3)} }$

$\implies \sf{ \dfrac{ {2}^{1999} }{ {2}^{1998} } \times \dfrac{5}{3} }$

We know that

$\dfrac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n}$

Applying the above law of exponent,

$\implies \sf{( {2}^{1999 - 1998} ) \times \dfrac{5}{3} }$

$\implies \sf{2 \times \dfrac{5}{3} }$

$\implies \sf{ \dfrac{10}{3} }$

$\therefore \: \: \boxed{ \bf{\dfrac{ {2}^{2001} + {2}^{1999} }{ {2}^{2000} - {2}^{1998} } = \dfrac{10}{ 3} }}$