How do you add or subtract #x^3/(x^2-y^2 )+ (y^3)/( y^2-x^2)#?

Question

Isaiah Anthony · Accepted Answer

We just need to find the lowest common denominator among these two fractions. However, we can easily see that one denominator is the opposite of the other (which means one is the "negative" of the other). This becomes clearer when we rewrite the operation:

#x^3/(x^2-y^2)+y^3/((-1)(x^2-y^2))#

So, now, we can consider as our lowest common denominator #(-1)(x^2-y^2)#.

Now, operating our sum...

#((-1)x^3+y^3)/((-1)(x^2-y^2))=(y^3-x^3)/(y^2-x^2)#

Nathan Axel · Answer

undefined To add or subtract the given expression, we need to find a common denominator. In this case, the common denominator is (x^2 - y^2)(y^2 - x^2).

To add the fractions, we multiply the numerator and denominator of the first fraction by (y^2 - x^2), and the numerator and denominator of the second fraction by (x^2 - y^2). This gives us:

(x^3(y^2 - x^2))/( (x^2 - y^2)(y^2 - x^2) ) + (y^3(x^2 - y^2))/( (x^2 - y^2)(y^2 - x^2) )

Simplifying the numerators, we have:

(x^3y^2 - x^5)/( (x^2 - y^2)(y^2 - x^2) ) + (x^2y^3 - y^5)/( (x^2 - y^2)(y^2 - x^2) )

Now, we can combine the fractions by adding the numerators:

(x^3y^2 - x^5 + x^2y^3 - y^5)/( (x^2 - y^2)(y^2 - x^2) )

Finally, we can simplify the numerator if needed, but we cannot further simplify the expression since the denominator cannot be factored.