Final answer:
To rationalize the denominator -11x^3/√(8x-4), we multiply both numerator and denominator by the conjugate of the denominator, leading to the simplified form -11x^3/2√(2x) + 1, which is option D.
Step-by-step explanation:
The student is asking to rationalize the denominator of the expression -11x^3/√(8x-4). To rationalize the denominator, we need to eliminate the square root from the denominator. We can do this by multiplying the numerator and the denominator by the conjugate of the denominator, which is √(8x-4) + √(8x-4). The conjugate of a binomial a + b is a - b, and it is used here to create a difference of squares when multiplied, which eliminates the square root.
First, express the denominator √(8x-4) as √(8x) - √(4) to simplify it, which is √(4 × 2x) - √(2 × 2), or 2√(2x) - 2. Now, the expression becomes -11x^3/(2√(2x) - 2). We multiply both the numerator and denominator by the conjugate of the denominator, (2√(2x) + 2), to rationalize.
After the multiplication, the denominator becomes 4 × 2x - 4, which simplifies to 8x - 4, and the expression simplifies to -11x^3/(8x-4). However, we must check if we can simplify it further. We find that the term 8x-4 is already factorized out and appears in both numerator and denominator, thus we cancel it out, leading to the simplified form -11x^3/2√(2x) + 1. Therefore the correct answer is D. -11x^3/2√(2x) + 1.