How do you evaluate #\frac { 2k ^ { 2} + 7k p + 6p ^ { 2} } { 9k ^ { 2} - 33k p + 28p ^ { 2} } \div \frac { 6k ^ { 2} + 17k p + 12p ^ { 2} } { 9k ^ { 2} - 16p ^ { 2} }#?

1 Answer
NJ
Mar 23, 2018

#=>(k+2p)/(3k-7p)#

Explanation:

#\frac { 2k ^ { 2} + 7k p + 6p ^ { 2} } { 9k ^ { 2} - 33k p + 28p ^ { 2} } \div \frac { 6k ^ { 2} + 17k p + 12p ^ { 2} } { 9k ^ { 2} - 16p ^ { 2} }#

First, let's invert the second fraction to be able to do multiplication.

#\frac { 2k ^ { 2} + 7k p + 6p ^ { 2} } { 9k ^ { 2} - 33k p + 28p ^ { 2} } * \frac { 9k ^ { 2} - 16p ^ { 2} } { 6k ^ { 2} + 17k p + 12p ^ { 2} }#

Now we have to do some factoring.

#((2 k + 3 p) (k + 2 p))/((3 k - 4 p) (3 k - 7 p)) * ( (3 k - 4 p) (3 k + 4 p))/((2 k + 3 p) (3 k + 4 p) )#

Now we can cancel some terms that divide out.

#(color(red)cancel{(2 k + 3 p)} (k + 2 p))/(color(blue)cancel{(3 k - 4 p)} (3 k - 7 p)) * ( color(blue)cancel{(3 k - 4 p)} color(orange)cancel{(3 k + 4 p)})/(color(red)cancel{(2 k + 3 p)} color(orange)cancel{(3 k + 4 p)} )#

And we're left with simply:

#=>(k+2p)/(3k-7p)#

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