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**: We can express 50 as \( 25 \times 2 \). 2. **Apply the square root**: Using the property of square roots, we have: \[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} \] 3. **Calculate \( \sqrt{25} \)**: Since \( \sqrt{25} = 5 \), we can substitute this back into our expression: \[ \sqrt{50} = 5\sqrt{2} \] ### Step 2: Simplifying \( \sqrt{18} \) 1. **Factor 18**: We can express 18 as \( 9 \times 2 \). 2. **Apply the square root**: Again, using the property of square roots: \[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} \] 3. **Calculate \( \sqrt{9} \)**: Since \( \sqrt{9} = 3 \), we substitute this back: \[ \sqrt{18} = 3\sqrt{2} \] ### 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 them together: \[ 5\sqrt{2} + 3\sqrt{2} = (5 + 3)\sqrt{2} = 8\sqrt{2} \] ### Final Answer Thus, the simplified form of the expression \( \sqrt{50} + \sqrt{18} \) is: \[ \boxed{8\sqrt{2}} \] ### Explanation of Options Now, let's analyze the provided options: - **Option A: \( 5\sqrt{2} + 3\sqrt{2} \)**: This is not the final answer but represents the intermediate step before combining like terms. It is correct in the sense that it shows the correct components, but it is not fully simplified. - **Option B: \( 4\sqrt{2} + 3\sqrt{3} \)**: This is incorrect because it misrepresents the simplification of \( \sqrt{50} \) and \( \sqrt{18} \). The terms do not match our calculations. - **Option C: \( 8\sqrt{2} \)**: This is the correct final answer. It accurately represents the simplified form of the original expression. - **Option D: \( 5\sqrt{5} + 3\sqrt{2} \)**: This is incorrect because it incorrectly simplifies \( \sqrt{50} \) as \( 5\sqrt{5} \), which is not valid. ### Common Pitfalls - **Not factoring correctly**: Always ensure you factor the numbers under the square root correctly to find perfect squares. - **Combining unlike terms**: Remember that you can only combine terms that are like terms (i.e., they must have the same radical part). - **Forgetting to simplify**: Always check if you can simplify the square roots before combining them. ### Revision Summary - Simplify square roots by factoring into perfect squares. - Combine like terms after simplification. - The final answer for \( \sqrt{50} + \sqrt{18} \) is \( 8\sqrt{2} \). - Be cautious of common mistakes in factoring and combining terms.