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36m5n5 ÷ (12m3)

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3 Answers

If I had to guess, I'd say the 5's and 3's are exponents? Working under this assumption:



A property of exponents is the following:

am ÷ an = a(m-n) 

Note that this is the inverse operation when multiplying like bases with exponents:

am x an = a(m+n)

This can be illustrated if we expand the factors with exponents in this way,



You can see that the three m's on the denominator can be cancelled with three on the numerator, leaving m2 on the numerator, proving that the property m(5-3) = m2

You can treat the coefficients separately, so that the ratio 36/12 is reduced to 3. The n3 factor is just along for the ride since it cannot be simplified further. 

So putting it all together we get: 



I'm assuming that you're meaning to write the following:


In this case, we compare similar bases' exponents. We subtract the lesser from the higher number and keep the rest wherever the higher number is.

More generally, if we have xn/xm, it would simplify to xn-m. You can see that in the case where m>n (n-m would be negative), then xn-m would be written as 1/(xm-n), if we want to keep positive exponents.

ANYWAY, in our case,


Let's first notice that 36/12 is equal to 3 = 3/1.


Now, using our exponent rule from above, we can see that we have an m and n in the numerator, and m in the denominator. Since m is the only one in common, that's the only one we need to worry about simplifying. The 5 on top is bigger than 3 on the bottom, so we know that m is going to end up on top. 5-3=2, so we have 2 m's left over on top.

Our answer would be:



My browser isn't playing nice with the edit button right now, so I just wanted to add that in the general case that I put - the m and n are separate from the problem and don't relate to our variables. Accidentally chose the exact same variables that we were working with!