Which expression is equivalent to the given expression? A. \( \frac{8 \sqrt{5}}{5} \) B. \( \frac{4 \sqrt{10}}{\sqrt{5}} \) C. \( \frac{4}{0} \sqrt{5} \) D. \( \sqrt{40} \)

To determine which expression is equivalent to the given expression, let's simplify each option and compare it to the original expression. The original expression is not provided, but I will assume it is something like \( \frac{8 \sqrt{5}}{5} \) since that is the first option and it seems to be the most likely candidate for equivalence. Let's simplify each option: A. \( \frac{8 \sqrt{5}}{5} \) - This is already in simplified form and matches the assumed original expression. B. \( \frac{4 \sqrt{10}}{\sqrt{5}} \) - To simplify, we can rationalize the denominator by multiplying the numerator and denominator by \( \sqrt{5} \): \[ \frac{4 \sqrt{10}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{4 \sqrt{10} \cdot \sqrt{5}}{5} = \frac{4 \sqrt{50}}{5} \] Since \( \sqrt{50} = \sqrt{25 \cdot 2} = 5 \sqrt{2} \), we have: \[ \frac{4 \cdot 5 \sqrt{2}}{5} = 4 \sqrt{2} \] This is not equivalent to the original expression. C. \( \frac{4}{0} \sqrt{5} \) - This expression is undefined because division by zero is not allowed. It cannot be equivalent to the original expression. D. \( \sqrt{40} \) - To simplify, we can factor 40 into its prime factors: \[ \sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10} \] This is not equivalent to the original expression. None of the options B, C, or D are equivalent to the original expression. However, option A is already in simplified form and matches the assumed original expression. Therefore, the correct answer is: A. \( \frac{8 \sqrt{5}}{5} \)