(p^2 - 2)/(p^2 - pq) To simplify this fraction, we can factor out a common factor from the numerator and the denominator: (p^2 - 2)/(p^2 - pq) = [(p + √2)(p - √2)]/[p(p - q)] Now we can cancel out the common factors: [(p + √2)(p - √2)]/[p(p - q)] = (p + √2)/(p - q)
(q^2 - 2)/(pq - p^2) To simplify this fraction, we can factor out a common factor from the numerator and the denominator: (q^2 - 2)/(pq - p^2) = (q^2 - 2)/[-p(p - q)] Now we can simplify the fraction by factoring out a negative sign from the denominator: (q^2 - 2)/[-p(p - q)] = -(q^2 - 2)/(p(p - q)) Now we can simplify further: -(q^2 - 2)/(p(p - q)) = (2 - q^2)/(p(q - p))
Now, we can add the two simplified fractions: (p + √2)/(p - q) + (2 - q^2)/(p(q - p))
This is the final simplified expression for the given expression.
First, let's simplify each fraction separately:
(p^2 - 2)/(p^2 - pq)
To simplify this fraction, we can factor out a common factor from the numerator and the denominator:
(p^2 - 2)/(p^2 - pq) = [(p + √2)(p - √2)]/[p(p - q)]
Now we can cancel out the common factors:
[(p + √2)(p - √2)]/[p(p - q)] = (p + √2)/(p - q)
(q^2 - 2)/(pq - p^2)
To simplify this fraction, we can factor out a common factor from the numerator and the denominator:
(q^2 - 2)/(pq - p^2) = (q^2 - 2)/[-p(p - q)]
Now we can simplify the fraction by factoring out a negative sign from the denominator:
(q^2 - 2)/[-p(p - q)] = -(q^2 - 2)/(p(p - q))
Now we can simplify further:
-(q^2 - 2)/(p(p - q)) = (2 - q^2)/(p(q - p))
Now, we can add the two simplified fractions:
(p + √2)/(p - q) + (2 - q^2)/(p(q - p))
This is the final simplified expression for the given expression.