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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:

36m5n3
_________

12m3

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,

m*m*m*m*m
_____________

m*m*m

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: 

3m2n3

Comments

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

(36m5n5)/(12m3)

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,

(36m5n5)/(12m3)

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

(3m5n5)/(m3)

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:

3m2n5

Comments

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!