What is the square root of (8 + 2√15)?

Step-by-Step Solution

To find the square root of (8 + 2 * sqrt(15)), we look for an expression in the form of (sqrt(a) + sqrt(b)) whose square equals (8 + 2 * sqrt(15)).

1. Use the algebraic identity: The square of (sqrt(a) + sqrt(b)) is (a + b + 2 * sqrt(ab)). Comparing this to (8 + 2 * sqrt(15)), we need to find two numbers, a and b, such that:

   • a + b = 8

   • a * b = 15

2. Find the values for a and b: The factors of 15 are (1, 15) and (3, 5). Testing the factors:

   • 3 + 5 = 8 (This matches our required sum)

   • 3 * 5 = 15 (This matches our required product)

3. Form the square root: So, the expression (8 + 2 * sqrt(15)) can be written as: (sqrt(5) + sqrt(3))^2 Therefore, the square root of (8 + 2 * sqrt(15)) is sqrt(5) + sqrt(3).