To simplify the expression \( \frac{2\sqrt{8}}{\sqrt{2}} \) and find the simplest form, let's go through the steps carefully. ### Step 1: Simplify the numerator First, we need to simplify \( \sqrt{8} \). We can break it down as follows: \[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \] Now, substituting this back into the expression, we have: \[ \frac{2\sqrt{8}}{\sqrt{2}} = \frac{2 \times 2\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{\sqrt{2}} \] ### Step 2: Simplify the fraction Next, we can simplify \( \frac{4\sqrt{2}}{\sqrt{2}} \). Since \( \sqrt{2} \) in the numerator and denominator are the same, they cancel each other out: \[ \frac{4\sqrt{2}}{\sqrt{2}} = 4 \] ### Step 3: Final answer Thus, the simplest form of the expression \( \frac{2\sqrt{8}}{\sqrt{2}} \) is: \[ 4 \] ### Conclusion The correct option is **A. \( 4 \)**. ### Explanation of Other Options - **B. \( 2\sqrt{2} \)**: This option is incorrect because it suggests that the expression simplifies to a value that is not equal to \( 4 \). The simplification process shows that the expression reduces to \( 4 \), not \( 2\sqrt{2} \). - **C. \( 4\sqrt{2} \)**: This option is also incorrect. It implies that the expression retains a radical component, which is not the case after simplification. The expression simplifies completely to a whole number, \( 4 \). - **D. \( \sqrt{2} \)**: This option is incorrect as well. It suggests that the expression simplifies to a smaller radical, which is not true. The simplification process shows that the expression does not retain any radicals. ### Common Pitfalls 1. **Not simplifying radicals**: Students often forget to simplify square roots fully, which can lead to incorrect answers. 2. **Misunderstanding cancellation**: It's crucial to recognize when terms in the numerator and denominator can be canceled out. 3. **Overlooking the final simplification**: After simplifying, always double-check the final result to ensure it is in its simplest form. ### Revision Summary - Simplify square roots fully before substituting back into the expression. - Cancel out common terms in the numerator and denominator. - Always verify the final answer to ensure it is in the simplest form. - The correct answer for \( \frac{2\sqrt{8}}{\sqrt{2}} \) is \( 4 \).