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Even if we change into
, the expression is equivalent to the first expression. So, this expression is symmetric.
In this case, we can evaluate with the sum and product.
As we know that -
and, -
we have to evaluate -
Let's rationalize -
And then -
Now we know that -
The given fraction is -
Hence, the answer is -
Answer:
63/61
Step-by-step explanation:
x = (√5 - √3)/(√5 + √3)
Rationalising the denominator,
x = (√5 - √3)/(√5 + √3) × (√5 - √3)/(√5 - √3)
x = [√5(√5 - √3) - √3(√5 - √3)]/(5 - 3)
x = (5 - √15 - √15 + 3)/2
x = (8 - 2√15)/2
x = 4 - √15
y = (√5 + √3)/(√5 - √3)
Rationalising the denominator,
y = (√5 + √3)/(√5 - √3) × (√5 + √3)/(√5 +√3)
y = [√5(√5 + √3) + √3(√5 + √3)]/(5 - 3)
y = (5 + √15 + √15 + 3)/2
y = (8 + 2√15)/2
y = 4 + √15
Now,
→ (x² + xy + y²)/(x² - xy + y²)
Used identity: (a + b)² = a² + b² + 2ab
So, we can write (x² + xy + y²)/(x² - xy + y²) as (x² + y² + 2xy - xy)/(x² + y² + 2xy - 3xy) or [(x + y)² - xy]/[(x + y)² - 3xy].
→ [(x + y)² - xy]/[(x + y)² - 3xy]
Substitute the values,
→ [(4 - √15 + 4 + √15)² - (4 - √15)(4 + √15)]/[(4 - √15 + 4 + √15)² - 3(4 - √15)(4 + √15)]
→ [(8)² - (16 + 4√15 - 4√15 - 15)]/[(8)² - 3(16 - 15)]
→ (64 - 1)/(64 - 3)
→ 63/61
Hence, the value of (x² + xy + y²)/(x² - xy + y²) is 63/61.