Bunuel wrote:
What is the value of 2^x + 2^(-x) ?
(1) x < 0
(2) 4^x + 4^(−x) = 23
NEW question from GMAT® Quantitative Review 2019
(DS05989)
OA: B
What is the value of \(2^x + 2^{-x}\) ?
or What is value of \(x\)?
(1) : \(x < 0\)
plugging \(x =-1\)
\(2^x + 2^{-x} = 2^{-1}+2^{1}= 0.5 + 2 =2.5\)
plugging \(x =-2\)
\(2^x + 2^{-x} = 2^{-2}+2^{2}= \frac{1}{4} + 4 = 0.25 + 4 = 4.25\)
We are not getting a unique value of \(2^x + 2^{-x}\)
So Statement 1 alone is not sufficient
(2) \(4^x + 4^{−x} = 23\)
Let \(2^x + 2^{-x}=y\)
Squaring both sides, we get
\((2^x + 2^{-x})^2=y^2\)
\((2^x)^2 +(2^{-x})^2 + 2(2^x)(2^{-x}) =y^2\)
\(2^x.2^x +2^{-x}.2^{-x} + 2 =y^2\)
\(4^x + 4^{−x} + 2 =y^2\)
\(23 + 2 =y^2\)
\(y^2=25 ; y=+5,-5\) (\(-5\) rejected as minimum value of \(2^x + 2^{-x}\) is \(2\) at \(x =0\))
using A.M≥ G.M
\(\frac{2^x + 2^{-x}}{2}≥\sqrt{{2^x * 2^{-x}}}\)
\(2^x + 2^{-x}≥2\)
So Statement 2 alone is sufficient