Simplify the following:

25*p*^{3}*q*^{2}

––––––

35*p**q*

(*r* – *s*)(2r + s)^{2}

–––––––––––

(*r* + *s*)(2r + s)

We can write out numerator and denominator as the products of their smallest factors:

25*p*^{3}*q*^{2}
5 × 5 × *p* × *p* × *p* × *q* × *q*
5 × *p* × *p* × *q*
5*p*^{2}*q*

–––––– =
––––––––––––––––––––
= ––––––––––
= –––––

35*p**q*
5 × 7 × *p* × *q*
7
7

We can cancel one 5, one *p*, and one *q* from top and bottom.
We then remove the × signs and make *p* × *p* into *p*^{2}.

(*r* – *s*)(2*r* + *s*)^{2}
(*r* – *s*)(2*r* + *s*)

–––––––––––
= –––––––––––

(*r* + *s*)(2*r* + *s*)
(*r* + *s*)

The only common factor in numerator and denominaotor is (2*r* + *s*).
We can cancel one of these on the bottom and one of the two on the top.
No more cancelling is possible,
because the remaining three terms are all different and have no common factor.

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Last updated: 26 November, 2007.