Question

Biết \({4^\alpha } + {4^{ - \alpha }} = 5\).

Tính giá trị của các biểu thức:

a) \({2^\alpha } + {2^{ - \alpha }}\);

b) \({4^{2\alpha }} + {4^{ - 2\alpha }}\).

 

Answer 1

a, Ta có: 

\(\left(2^{\alpha}+2^{-\alpha}\right)^2\\ =\left(2^{\alpha}\right)^2+2\cdot2^{\alpha}\cdot2^{-\alpha}+\left(2^{-\alpha}\right)^2\\ =4^{\alpha}+4^{-\alpha}+2\\ =5+2\\ =7\)

Vậy \(2^{\alpha}+2^{-\alpha}=7\)

b, Ta có: 

\(4^{2\alpha}+4^{-2\alpha}=\left(4^{\alpha}\right)^2+\left(4^{-\alpha}\right)^2\\ =\left(4^{\alpha}+4^{-\alpha}\right)^2-2\\ =5^2-2\\ =23\)

Vậy \(4^{2\alpha}+4^{-2\alpha}=23\)