Solve the equation 1/2^(x^2-1)=square root(16^x).

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We have to solve the equation 1/2^(x^2-1) = sqrt (16^x)

1/2^(x^2-1) = sqrt (16^x)

=> 2^-(x^2 - 1) = 16^x^(1/2)

=> 2^(-x^2 + 1) = 2^4^x^(1/2)

=> 2^(-x^2 + 1) = 2^2x

We can equate the exponent as the base is the same

-x^2 + 1 = 2x

=> x^2 + 2x - 1 = 0

x1 = [-2 + sqrt (4 + 4)]/2

=> x1 = -1 + sqrt 2

x2 = -1 - sqrt 2

The required solution is x = -1 + sqrt 2 and x = -1 - sqrt 2.

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