Indices - factorising

Question

Simplify: $$\frac {x^5y^2x^3 + x^4y^5 - y^5x^7y^4}{x^4y^3}$$

I know this is probably low level stuff but I need to be able to do this specific type of question and I have no way of checking my work. If anyone could offer a step by step working I'd be appreciative

Answer 1

You have \begin{align*} \frac {x^5y^2x^3 + x^4y^5 - y^5x^7y^4}{x^4y^3} &=\frac {x^8y^2 + x^4y^5 - x^7y^9}{x^4y^3} \\ &=\frac {x^4y^2\big(x^4 + y^3 - x^3y^7\big)}{x^4y^3} \\ &=\frac {x^4 + y^3 - x^3y^7}{y},\quad x\not=0. \end{align*} That's about as far as you can go.

Answer 2

The highest common factor of the terms in the numerator is $x^4y^2.$ This is also a factor of the denominator, hence you can cancel it off to get $$\frac{x^4+y^3-x^3y^7}{y}.$$

Answer 3

An alternate way is to break the problem down into simple workable parts: $$\frac{x^5y^2x^3+x^4y^5−y^5x^7y^4}{x^4y^3}$$ $$=\frac{x^5y^2x^3}{x^4y^3}+\frac{x^4y^5}{x^4y^3}-\frac{y^5x^7y^4}{x^4y^3}$$ $$=\frac{x^8y^2}{x^4y^3}+\frac{x^4y^5}{x^4y^3}-\frac{x^7y^9}{x^4y^3}$$ $$=\frac{x^4}{y}+y^2-x^3y^6,\quad x\not=0.$$ This is equivalent to the answers given above.