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Sagot :
Answer:
Es igual a 7.
Step-by-step explanation:
Ok, sabemos que:
[tex]2^x + 2^{-x} = 3[/tex]
Queremos calcular:
[tex]4^x + 4^{-x}[/tex]
Tambien sabemos que 4 = 2*2
Entonces:
[tex]4^x + 4^{-x} = (2*2)^x + (2*2)^{-x}[/tex]
y sabemos que 2*2 = 2^2
Entonces:
[tex]4^x + 4^{-x} = (2*2)^x + (2*2)^{-x} = (2^2)^x + (2^2)^{-x}[/tex]
y ahora podemos usar la relación:
[tex](a^n)^m = (a^m)^n = a^{m*n}[/tex]
entonces:
[tex](2^2)^x + (2^2)^{-x} = (2^x)^2 + (2^{-x})^2[/tex]
Ahora podemos completar cuadrados sumando y restando el termino:
[tex]2*(2^x)*(2^{-x})[/tex]
Asi tendremos:
[tex](2^x)^2 + (2^{-x})^2 = (2^x)^2 + (2^{-x})^2 + 2*(2^x)*(2^{-x}) - 2*(2^x)*(2^{-x}) = (2^x + 2^{-x})^2 - 2*(2^x)*(2^{-x})[/tex]
Entonces de momento tenemos que:
[tex]4^x + 4^{-x} = (2^x + 2^{-x})^2 - 2*(2^x)*(2^{-x})[/tex]
Sabemos que el termino que esta dentro del parentesis es igual a 3.
Y tambien podemos usar la propiedad:
[tex]a^n*a^m = a^{n + m}[/tex]
en el termino de la derecha.
Asi tendremos:
[tex](2^x + 2^{-x})^2 - 2*(2^x)*(2^{-x}) = (3)^2 - 2*2^{x + (-x)} = 9 - 2 = 7[/tex]
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