Answer: To simplify the given expression, we can start by noticing that we have a difference of squares in each part of the expression. Recall the difference of squares formula:
a^2 - b^2 = (a + b)(a - b)
Now, let's simplify step by step:
In the first part of the expression, we have (3 − √(5)) × √(3 + √(5)). This part can be rewritten as a difference of squares:
(3 − √(5)) × √(3 + √(5)) = (3 - √5) × √(3 + √5) × (√(3 + √5) + √(3 - √5)) / (√(3 + √5) + √(3 - √5))
The denominator (√(3 + √5) + √(3 - √5)) is a conjugate of the numerator (√(3 + √5) - √(3 - √5)), and when multiplied, it results in a difference of squares:
(3 - √5) × √(3 + √5) × (√(3 + √5) + √(3 - √5)) / (√(3 + √5) + √(3 - √5)) = (3 + √5) × (3 + √5) = (3 + √5)^2
In the second part of the expression, we have (3 + √(5)) × √(3 − √(5)). Similarly, this can be rewritten as a difference of squares:
(3 + √(5)) × √(3 − √(5)) = (3 + √5) × √(3 − √5) × (√(3 + √5) - √(3 - √5)) / (√(3 + √5) - √(3 - √5))
The denominator (√(3 + √5) - √(3 - √5)) is a conjugate of the numerator (√(3 + √5) + √(3 - √5)), and when multiplied, it results in a difference of squares:
(3 + √5) × √(3 − √5) × (√(3 + √5) - √(3 - √5)) / (√(3 + √5) - √(3 - √5)) = (3 - √5) × (3 - √5) = (3 - √5)^2
Now, let's put both parts together:
(3 − √(5)) × √(3 + √(5)) + (3 + √(5)) × √(3 − √(5)) = (3 + √5)^2 + (3 - √5)^2
Simplify each square:
(3 + √5)^2 = 3^2 + 2 * 3 * √5 + (√5)^2 = 9 + 6√5 + 5 = 14 + 6√5
(3 - √5)^2 = 3^2 - 2 * 3 * √5 + (√5)^2 = 9 - 6√5 + 5 = 14 - 6√5
Add the simplified squares together:
(3 + √5)^2 + (3 - √5)^2 = (14 + 6√5) + (14 - 6√5) = 28
So, the simplified expression is 28.