To simplify the expression \( \frac{3\sqrt{8}}{\sqrt{2}} \), we will follow a step-by-step approach. ### Step 1: Simplify the square roots First, we need to simplify \( \sqrt{8} \). We can express \( 8 \) as \( 4 \times 2 \): \[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \] ### Step 2: Substitute back into the expression Now, we can substitute \( \sqrt{8} \) back into the original expression: \[ \frac{3\sqrt{8}}{\sqrt{2}} = \frac{3 \times 2\sqrt{2}}{\sqrt{2}} \] ### Step 3: Simplify the fraction Next, we can simplify the fraction. The \( \sqrt{2} \) in the numerator and the denominator can be canceled out: \[ \frac{3 \times 2\sqrt{2}}{\sqrt{2}} = 3 \times 2 = 6 \] ### Final Answer Thus, the simplified form of the expression \( \frac{3\sqrt{8}}{\sqrt{2}} \) is: \[ \boxed{6} \] ### Explanation of Other Options - **Option A**: [blank] - This option does not provide any value, so it cannot be correct. - **Option C**: [blank] - Similar to Option A, this option is also blank and cannot be correct. - **Option D**: \( 3\sqrt{4} \) - While \( \sqrt{4} = 2 \), thus \( 3\sqrt{4} = 3 \times 2 = 6 \), this option is not in its simplest form compared to the direct answer of \( 6 \). Therefore, while it is mathematically equivalent, it is not the simplest representation. ### Summary of Key Points - To simplify square roots, factor them into perfect squares when possible. - Always simplify fractions by canceling common terms in the numerator and denominator. - The final answer for \( \frac{3\sqrt{8}}{\sqrt{2}} \) is \( 6 \). - Be cautious of options that may seem equivalent but are not in their simplest form. This thorough approach ensures a clear understanding of how to simplify expressions involving square roots and fractions.