Simplify \( \frac{8+\sqrt{6}}{\sqrt{2}} \) \( \frac{\sqrt{2}+4 \sqrt{3}}{4 \sqrt{6}} \) \( 4 \sqrt{2}+\sqrt{3} \)

We are given three expressions to simplify. One acceptable answer for each is given below. ────────────────────────────── 1) Simplify (8 + √6) / √2 Step 1. Write the expression as a sum:   (8 + √6) / √2 = 8/√2 + √6/√2 Step 2. Simplify each term:   8/√2 = 8·(√2/2) = 4√2  (since 8/√2 = (8√2)/(√2·√2) = 8√2/2)   √6/√2 = √(6/2) = √3 Thus,   (8+√6)/√2 = 4√2 + √3 ────────────────────────────── 2) Simplify (√2 + 4√3) / (4√6) Step 1. Split the fraction:   (√2 + 4√3)/(4√6) = √2/(4√6) + 4√3/(4√6) = (√2/(4√6)) + (√3/√6) Step 2. Simplify each term separately. For the first term:   √2/(4√6)   Multiply numerator and denominator by √6:     = (√2·√6)/(4·6) = √12/24   Since √12 = 2√3 we have:     = (2√3)/24 = √3/12 For the second term:   √3/√6 = √(3/6) = √(1/2) = 1/√2   Alternatively, you can rationalize this:     1/√2 = √2/2 Now we have:   (√2 + 4√3)/(4√6) = (√3/12) + (√2/2) Step 3. Write with a common denominator:   Express √2/2 with denominator 12:     √2/2 = (6√2)/12 Thus,   = (√3 + 6√2) / 12 ────────────────────────────── 3) Simplify 4√2 + √3 This expression is already in simplest form, so no further simplification is necessary. ────────────────────────────── Summary of Answers 1) (8 + √6) / √2 = 4√2 + √3 2) (√2 + 4√3) / (4√6) = (6√2 + √3) / 12 3) 4√2 + √3 is already in simplest form. Any of these forms is acceptable unless a specific format is required.