Find the value of ( x³ + y³ ) if :..........?

Let's see what we can do with this:

x = (√7 - √5) / (√7 + √5) and y = (√7 + √5) / (√7 - √5)

Before moving on, let's simplify this by rationalizing the denominators:

x = (√7 - √5)² / [(√7 + √5)(√7 - √5)] and y = (√7 + √5)² / [(√7 + √5)(√7 - √5)]
x = (√7 - √5)² / (7 - √35 + √35 - 5) and y = (√7 + √5)² / (7 - √35 + √35 - 5)
x = (√7 - √5)² / (7 - 5) and y = (√7 + √5)² / (7 - 5)
x = (√7 - √5)² / 2 and y = (√7 + √5)² / 2
x = (7 - 2√35 + 5) / 2 and y = (7 + 2√35 + 5) / 2
x = (12 - 2√35) / 2 and y = (12 + 2√35) / 2
x = 6 - √35 and y = 6 + √35

Knowing this, you want the value of:

x³ + y³

Let's factor that, first:

(x + y)(x² - xy + y²)

Now if we substitute values, we get:

(6 - √35 + 6 + √35)[(6 - √35)² - (6 - √35)(6 + √35) + (6 + √35)²]

Now simplify what's left:

12[(36 - 12√35 + 35) - (36 + 6√35 - 6√35 - 35) + (36 + 12√35 + 35)]
12[(71 - 12√35) - (36 - 35) + (71 + 12√35)]
12(71 - 12√35 - 1 + 71 + 12√35)
12(71 - 1 + 71)
12(141)
1692

The answer is as follows:

x + y
= [(√7 - √5)/(√7 + √5)] + [(√7 + √5)/(√7 - √5)]
= [(√7 - √5)^2 + (√7 + √5)^2]/[(√7 + √5)(√7 - √5)]
= [7 - 2√35 + 5) + (7 + 2√35 + 5)]/[7 - 5]
= 24/2
= 12

xy
= [(√7 - √5)/(√7 + √5)][(√7 + √5)/(√7 - √5)]
= [(√7 - √5)(√7 + √5)]/[(√7 + √5)(√7 - √5)]
= 1

So
x^3 + y^3
= (x + y)(x^2 - xy + y^2)
= (x + y)[(x + y)^2 - 3xy]
= 12[12^2 - 3*1]
= 12[141]
= 1692