Final answer:
To simplify (9x^2y^4z^8)^(-1/2), we invert the square root of the expression, leading to the answer C) 1/3xy^2z^4. There is no negative value in this simplified expression.
Step-by-step explanation:
To simplify the expression (9x^2y^4z^8)^(-1/2), we must understand that a negative exponent indicates that the term is inverted, and the fractional exponent represents a root. So, (-1/2) as the exponent means we take the reciprocal of the square root of the term. Here's the step-by-step simplification:
Firstly, remember that the square root of a product is the product of the square roots. Therefore, we have:
√(9x^2y^4z^8)^(-1) = (1/(√9)) × (1/(√(x^2))) × (1/(√y^4)) × (1/(√z^8)) = 1/(3xy^2z^4)
So, the simplified result from the given options is C) 1/3xy^2z^4.
Whenever we encounter negative exponents, the base of the exponent is taken to the denominator, and the fractional exponent indicates which root of the base is taken. No number or variable in the expression is negative, hence we do not end with a negative result.