To simplify the expression \( \sqrt{50} + \sqrt{18} \), we will break down each square root into its simplest form. Let's go through the steps carefully. ### Step 1: Simplifying \( \sqrt{50} \) 1. **Factor 50**: \[ 50 = 25 \times 2 \] Here, 25 is a perfect square. 2. **Apply the square root**: \[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \] ### Step 2: Simplifying \( \sqrt{18} \) 1. **Factor 18**: \[ 18 = 9 \times 2 \] Again, 9 is a perfect square. 2. **Apply the square root**: \[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \] ### Step 3: Combine the simplified forms Now that we have simplified both square roots, we can combine them: \[ \sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} \] ### Step 4: Combine like terms Since both terms contain \( \sqrt{2} \), we can add the coefficients: \[ 5\sqrt{2} + 3\sqrt{2} = (5 + 3)\sqrt{2} = 8\sqrt{2} \] ### Final Answer The simplified form of the expression \( \sqrt{50} + \sqrt{18} \) is: \[ 8\sqrt{2} \] ### Evaluating the Options Now, let's evaluate the provided options: - **Option A: \( 5\sqrt{2} + 3\sqrt{2} \)** This is not the final simplified form; it is an intermediate step. While it is correct in terms of the components, it does not represent the final answer. - **Option B: \( 4\sqrt{2} + 2\sqrt{3} \)** This option is incorrect because it does not match our simplified expression. The coefficients do not correspond to the simplifications we performed. - **Option C: \( 7\sqrt{2} \)** This option is incorrect as it does not reflect the correct sum of the coefficients. We found \( 8\sqrt{2} \). - **Option D: \( 5\sqrt{2} + 3\sqrt{3} \)** This option is incorrect because it incorrectly combines the square roots. The term \( 3\sqrt{3} \) does not appear in our simplification. ### Conclusion The correct answer is not listed among the options provided. The final simplified form of \( \sqrt{50} + \sqrt{18} \) is \( 8\sqrt{2} \). ### Revision Summary - To simplify square roots, factor them into perfect squares. - Combine like terms when they share the same radical. - Always check if the final answer matches the simplified form. - Be cautious of options that may seem correct but do not represent the final simplified expression.