Introduction

To simplify rational expressions, expand the powers and then cancel.

For example, $a^2/a^3=(a*a)/(a*a*a)=1/a$. In this example, we cancelled a 2 times, in both the numerator and denominator.

Another example, $a^2/a^4=(a*a)/(a*a*a*a)=1/(a*a)=1/a^2$

Another example, $a^4.y^2/x^2.y=(x*x*x*x*y*y)/(x*x*y)=x*x*y=x^2.y$. In this example, the denominator is 1, but dividing anything by 1 just gives itself.

If we had $(a^3+a^4)/a^3$, note we now have a plus sign (rather than just multiply). To solve this, we could factor out the $a^3$ from the numerator (top) and then cancel. This would give us $(a^3[1+a])/a^3$, and cancelling out the $a^3$, we have $1+a$.

Alternatively, we can split the quotient into 2 parts before cancelling. That is, $(a^3+a^4)/a^3=a^3/a^3+a^4/a^3=1+a$.

I find the 2nd method easier.