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irina1246 [14]
3 years ago
13

Ayudemen porfavor . Determine 4 ala potencia x + 4 ala potencia -x, si se sabe que 2ala potencia x + 2 ala potencia -x =3

Mathematics
1 answer:
Afina-wow [57]3 years ago
7 0

Answer:

Es igual a 7.

Step-by-step explanation:

Ok, sabemos que:

2^x + 2^{-x} = 3

Queremos calcular:

4^x + 4^{-x}

Tambien sabemos que 4 = 2*2

Entonces:

4^x + 4^{-x} = (2*2)^x + (2*2)^{-x}

y sabemos que 2*2 = 2^2

Entonces:

4^x + 4^{-x} = (2*2)^x + (2*2)^{-x} = (2^2)^x + (2^2)^{-x}

y ahora podemos usar la relación:

(a^n)^m = (a^m)^n = a^{m*n}

entonces:

(2^2)^x + (2^2)^{-x} = (2^x)^2 + (2^{-x})^2

Ahora podemos completar cuadrados sumando y restando el termino:

2*(2^x)*(2^{-x})

Asi tendremos:

(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})

Entonces de momento tenemos que:

4^x + 4^{-x} = (2^x + 2^{-x})^2 - 2*(2^x)*(2^{-x})

Sabemos que el termino que esta dentro del parentesis es igual a 3.

Y tambien podemos usar la propiedad:

a^n*a^m = a^{n + m}

en el termino de la derecha.

Asi tendremos:

(2^x + 2^{-x})^2 - 2*(2^x)*(2^{-x}) = (3)^2 - 2*2^{x + (-x)}  = 9 - 2 = 7

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