Find the highest value among the following

Correct Option
Option C: √5 + √10

Step-by-Step Solution
To compare the sums of square roots, we square each expression. Since all terms are positive, if a² > b², then a > b. The formula for squaring these expressions is (√x + √y)² = x + y + 2√(xy).

Option A: (√8 + √5)²

8 + 5 + 2√(8 * 5) = 13 + 2√40 ≈ 13 + 2(6.32) = 13 + 12.64 = 25.64

Option B: (√6 + √7)²

6 + 7 + 2√(6 * 7) = 13 + 2√42 ≈ 13 + 2(6.48) = 13 + 12.96 = 25.96

Option C: (√5 + √10)²

5 + 10 + 2√(5 * 10) = 15 + 2√50 ≈ 15 + 2(7.07) = 15 + 14.14 = 29.14

Option D: (√2 + √11)²

2 + 11 + 2√(2 * 11) = 13 + 2√22 ≈ 13 + 2(4.69) = 13 + 9.38 = 22.38

Comparing the results (25.64, 25.96, 29.14, and 22.38), 29.14 is the highest value.

Summary Rule
When comparing expressions of the form √x + √y, squaring the expression using (√x + √y)² = x + y + 2√(xy) allows you to transform the comparison into a simpler form where you look for the largest sum of x+y and 2√(xy).