Final answer:
The expression -1/2 · √(12x) simplifies to -√(3x) after factoring out the perfect square from 12 and simplifying the multiplication.
Step-by-step explanation:
When asked to rewrite the expression -1/2 · √(12x) as a square root or its opposite, we first need to manipulate the expression inside the square root. Since the square root of a product is the product of the square roots, we simplify the expression within the square root by factoring out perfect squares.
Beginning with √(12x), we recognize that 12 can be factored into 4 and 3, where 4 is a perfect square. Thus, √(12x) can be rewritten as √(4 × 3x), which simplifies to √(4) × √(3x), or 2√(3x). Therefore, the entire expression becomes -1/2 × 2√(3x).
Next, we can simplify by multiplying -1/2 by 2, which nullifies the fraction, leaving us with -√(3x). This is the simplified form of the original expression as a square root or its opposite.
Understanding properties such as negative exponents, taking square roots of exponentials, and the behavior of expressions under operations of multiplication and division is crucial to simplify square roots and rewrite expressions.