((y/(x-y)+(x+y)/y))/((x-y)/x+(y/(x+y)) can you help simplify please?

Answer 1

This is just an exercise in getting common denominators, adding fractions, and dividing fractions.

Working only with the numerator:

The common denominator is (x - y)(y). Multiply y/(x - y) by y/y and multiply (x + y)/y by (x - y)/(x - y).

It becomes:

y²/[(x - y)(y)] + [(x + y)(x - y)]/[(x - y)(y)]

y²/[(x - y)(y)] + (x² - y²)/[(x - y)(y)]

(y² + x² - y²)/[(x - y)(y)]

x²/[(x - y)(y)]

Working only with the denominator:

The common denominator is (x)(x + y). Multiply (x - y/x by (x + y)/(x + y) and multiply y/(x + y) by x/x.

It becomes:

[(x - y)(x + y)]/[(x)(x + y)] + xy/[(x)(x + y)]

(x² - y²)/[(x)(x + y)] + xy/[(x)(x + y)]

(x² + xy - y²)/[(x)(x + y)]

Now, you are dividing 2 fractions so keep the top as is, change the problem to multiplication, and flip the bottom one:

x²/[(x - y)(y)] • [(x)(x + y)]/(x² + xy - y²) which you can multiply out if desired:

[x³(x + y)]/[(x - y)(y)(x² + xy - y²)]

(x⁴ + x³y)/[(xy - y²)(x² + xy - y²)]

(x⁴ + x³y)/(x³y - 2xy³ - y⁴)

Answer 2

Usung LCD numerator simplifies to:

x² / (y(x - y))

Using LCD denominator simplifies to:

(x² + y² + xy) / (x(x + y))

Invert deniminator, multiply, and simplify.