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 also express 25 as \( 5^2 \). 2. **Rewrite the square root**: \[ \sqrt{50} = \sqrt{2 \times 25} = \sqrt{2} \times \sqrt{25} \] 3. **Simplify**: - 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 9 as \( 3^2 \). 2. **Rewrite the square root**: \[ \sqrt{18} = \sqrt{2 \times 9} = \sqrt{2} \times \sqrt{9} \] 3. **Simplify**: - Since \( \sqrt{9} = 3 \), we have: \[ \sqrt{18} = 3\sqrt{2} \] ### Step 3: Combine the simplified terms 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: \( 5\sqrt{2} + 3\sqrt{2} \)**: This is an intermediate step in the simplification process, not the final answer. While it is correct in the context of showing how we arrived at the final answer, it does not represent the complete simplification. - **Option B: \( 8\sqrt{2} \)**: This is the correct final answer, as shown in our calculations. - **Option C: \( 7\sqrt{2} \)**: This is incorrect because it miscalculates the sum of the coefficients. The correct sum is \( 5 + 3 = 8 \), not 7. - **Option D: \( 4\sqrt{3} \)**: This is incorrect as it does not relate to the original expression at all. The square roots of 50 and 18 do not simplify to involve \( \sqrt{3} \). ### Revision Summary - To simplify square roots, factor the numbers into their prime factors and identify perfect squares. - Combine like terms when adding square roots that share the same radical part. - Always check your calculations to ensure that you are adding the coefficients correctly. - The final answer for \( \sqrt{50} + \sqrt{18} \) is \( 8\sqrt{2} \).