To simplify the expression \((\sqrt{98} - \sqrt{50})/\sqrt{32}\), we will break it down step by step. ### Step 1: Simplify the square roots in the numerator 1. **Simplifying \(\sqrt{98}\)**: - We can factor \(98\) into \(49 \times 2\). - Since \(49\) is a perfect square, we can simplify: \[ \sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2} \] 2. **Simplifying \(\sqrt{50}\)**: - We can factor \(50\) into \(25 \times 2\). - Since \(25\) is a perfect square, we can simplify: \[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \] ### Step 2: Substitute back into the expression Now we can substitute the simplified square roots back into the original expression: \[ \frac{\sqrt{98} - \sqrt{50}}{\sqrt{32}} = \frac{7\sqrt{2} - 5\sqrt{2}}{\sqrt{32}} \] ### Step 3: Combine the terms in the numerator In the numerator, we can combine like terms: \[ 7\sqrt{2} - 5\sqrt{2} = (7 - 5)\sqrt{2} = 2\sqrt{2} \] ### Step 4: Simplify the denominator Next, we simplify \(\sqrt{32}\): - We can factor \(32\) into \(16 \times 2\). - Since \(16\) is a perfect square, we can simplify: \[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \] ### Step 5: Substitute back into the expression Now we substitute the simplified denominator back into the expression: \[ \frac{2\sqrt{2}}{4\sqrt{2}} \] ### Step 6: Cancel out the common terms We can cancel \(\sqrt{2}\) from the numerator and the denominator: \[ \frac{2}{4} = \frac{1}{2} \] ### Final Answer Thus, the simplified expression is: \[ \frac{1}{2} \] ### Explanation of Options - **Option A: 3** - This option is incorrect because the simplification does not yield a whole number; the calculations show that the result is a fraction. - **Option B: 1** - This option is incorrect as the simplification results in \(\frac{1}{2}\), not \(1\). - **Option C: \(\frac{1}{2}\)** - This is the correct answer as shown in the detailed steps above. - **Option D: \(\frac{1}{4}\)** - This option is incorrect because the simplification does not yield \(\frac{1}{4}\); the calculations confirm that the result is \(\frac{1}{2}\). ### Revision Summary - Simplify square roots by factoring out perfect squares. - Combine like terms in the numerator after simplification. - Simplify the denominator similarly and cancel common factors. - The final simplified result of \((\sqrt{98} - \sqrt{50})/\sqrt{32}\) is \(\frac{1}{2}\).