To simplify the expression \( \sqrt{50} + \sqrt{18} \), we will break down each square root into its simplest form and then combine like terms. ### Step 1: Simplifying \( \sqrt{50} \) 1. **Factor 50**: - The prime factorization of 50 is \( 2 \times 25 \). - We can express this as \( 50 = 2 \times 5^2 \). 2. **Apply the square root**: - Using the property of square roots, \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), we can write: \[ \sqrt{50} = \sqrt{2 \times 25} = \sqrt{2} \times \sqrt{25} \] - Since \( \sqrt{25} = 5 \), we have: \[ \sqrt{50} = 5\sqrt{2} \] ### Step 2: Simplifying \( \sqrt{18} \) 1. **Factor 18**: - The prime factorization of 18 is \( 2 \times 9 \). - We can express this as \( 18 = 2 \times 3^2 \). 2. **Apply the square root**: - Again using the property of square roots: \[ \sqrt{18} = \sqrt{2 \times 9} = \sqrt{2} \times \sqrt{9} \] - Since \( \sqrt{9} = 3 \), we have: \[ \sqrt{18} = 3\sqrt{2} \] ### Step 3: Combine the simplified square roots 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 Thus, the simplified form of the expression \( \sqrt{50} + \sqrt{18} \) is: \[ \boxed{8\sqrt{2}} \] ### Explanation of Options - **Option A: \( 7\sqrt{2} \)**: This is incorrect because it does not account for the correct addition of the coefficients from \( 5\sqrt{2} \) and \( 3\sqrt{2} \). - **Option B: \( 5\sqrt{2} + 3\sqrt{2} \)**: While this expression is correct in its form, it is not the final simplified answer. It represents the intermediate step before combining like terms. - **Option C: \( 4\sqrt{2} \)**: This is incorrect as it underestimates the sum of the coefficients from the two square roots. - **Option D: \( 8\sqrt{2} \)**: This is the correct final answer, as we have shown through the simplification process. ### Common Pitfalls - **Not simplifying square roots fully**: Always ensure to break down square roots into their prime factors to find the simplest form. - **Forgetting to combine like terms**: When adding square roots, ensure that you only add the coefficients of like terms (those with the same radical). - **Miscalculating coefficients**: Double-check the arithmetic when adding coefficients to avoid errors. ### Revision Summary - Simplify square roots by factoring into prime components. - Combine like terms carefully when adding square roots. - Always check your final answer against the options provided. - Understand the properties of square roots to avoid common mistakes.