Question

Se
${2}^{n} + {2}^{ - n} = 9$
Determine o valor de
A)
${4}^{n} + {4}^{ - n}$
B)
${8}^{n} + {8}^{ - n}$
Coloque a explicação pfv​

Answer 1

$\large\boxed{\begin{array}{l}\rm2^n+2^{-n}=9\\\sf a)~\rm 4^n+4^{-n}=(2^n)^2+(2^{-n})^2\\\rm (2^n+2^{-n})^2=(2^n)^2+2\cdot 2^n\cdot 2^{-n}+(2^{-n})^2\\\rm (2^n)^2+(2^{-n})^2=(2^n+2^{-n})^2-2\cdot 2^n\cdot 2^{-n}\\\rm (2^n)^2+(2^{-n})^2=9^2-2\cdot2^0\\\rm (2^n)^2+(2^{-n})^2=81-2\\\rm 4^n+4^{-n}=79\end{array}}$

$\large\boxed{\begin{array}{l}\sf c)~\rm 8^n+8^{-n}=(2^n)^3+(2^{-n})^3\\\rm a^3+b^3=(a+b)\cdot (a^2-ab+b^2)\\\rm (2^n)^3+(2^{-n})^3=(2^n+2^{-n})((2^n)^2-2^n\cdot 2^{-n}+(2^{-n})^2)\\\\\rm (2^n)^3+(2^{-n})^3=(2^n+2^{-n})\cdot( 4^n-1+4^{-n})\\\rm (2^n)^3+(2^{-n})^3=(2^n+2^{-n})(4^n+4^{-n}-1)\\\rm 8^n+8^{-n}=9\cdot(79-1)\\\rm 8^n+8^{-n}=9\cdot78\\\rm 8^n+8^{-n}=702\end{array}}$