To simplify the expression \(\sqrt{50} + \sqrt{18} - \sqrt{8}\), we will break down each square root into its simplest form. Let's go through this step-by-step. ### Step 1: Simplifying Each Square Root 1. **Simplifying \(\sqrt{50}\)**: - We can factor \(50\) into \(25 \times 2\). - Since \(25\) is a perfect square, we can take the square root of it: \[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \] 2. **Simplifying \(\sqrt{18}\)**: - We can factor \(18\) into \(9 \times 2\). - Again, \(9\) is a perfect square: \[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \] 3. **Simplifying \(\sqrt{8}\)**: - We can factor \(8\) into \(4 \times 2\). - Here, \(4\) is also a perfect square: \[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \] ### Step 2: Substituting Back into the Expression Now that we have simplified each square root, we can substitute them back into the original expression: \[ \sqrt{50} + \sqrt{18} - \sqrt{8} = 5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2} \] ### Step 3: Combining Like Terms Next, we combine the terms that contain \(\sqrt{2}\): - The coefficients of \(\sqrt{2}\) are \(5\), \(3\), and \(-2\): \[ 5 + 3 - 2 = 6 \] Thus, we have: \[ 5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2} = (5 + 3 - 2)\sqrt{2} = 6\sqrt{2} \] ### Final Expression The simplified form of the expression \(\sqrt{50} + \sqrt{18} - \sqrt{8}\) is: \[ 6\sqrt{2} \] ### Evaluating the Options Now, let's evaluate the provided options: - **Option A: \(5 + 3\sqrt{2}\)** - This is incorrect because it does not match our final result of \(6\sqrt{2}\). - **Option B: \(5 + 2\sqrt{2}\)** - This is also incorrect for the same reason. - **Option C: \(7 + 3\sqrt{2}\)** - This is incorrect as well. - **Option D: \(4 + 5\sqrt{2}\)** - This is incorrect too. None of the options provided match the correct simplified form of \(6\sqrt{2}\). ### Common Pitfalls - **Not recognizing perfect squares**: It's essential to identify perfect squares when simplifying square roots. - **Combining terms incorrectly**: Ensure that you only combine like terms (terms that have the same radical part). - **Forgetting to subtract**: When dealing with subtraction in expressions, be careful to apply it correctly to the coefficients. ### Revision Summary - Simplify square roots by factoring out perfect squares. - Combine like terms carefully, ensuring to account for addition and subtraction. - Always check your final answer against the options provided to ensure accuracy. - Practice simplifying various square root expressions to build confidence.