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Solution
What Rassoul did is correct, I am just going to offer a different layout approach and take the generic m’s and n’s away
whenever you have a variable as part of your exponent try to figure out what base all of the whole numbers can be simplified to. If all of the bases are the same you can kind of “forget” about the bases later. For this problem 216 and 36 have a base of 6 (6 can be raised to a whole exponent to achieve both 216 and 36)
63 = 216 and 62 = 36
Pretend the exponents are not there for a minute, and we can get:
216 x 36 = 36
63 x 62 = 6 2
Just take the exponents now and place those numbers in front of the original exponent in ( )
216 original exponent was (-r-1), new exponent is 3. Makes: 3(-r-1)
first 36 original exponent was (-3r-2), new exponent is 2. Makes: 2(-3r-2)
second 36 original exponent was (2r-1), new exponent is 2. Makes: 2(2r-1)
Put back into equation in their respective postitions, but don’t use the bases (6’s) anymore. This is the part where we get to “forget” about the bases.
3(-r-1) x 2(-3r-2) = 2(2r-1)
Remember that these are exponents, so exponent rules apply. When you multiply exponents you actually add them.
3(-r-1) + 2(-3r-2) = 2(2r-1)
Now solve as a regular algebra problem
-3r – 3 – 6r – 4 = 4r – 2
-9r – 7 = 4r – 2
13r = -5
r = -5/13
Same answer, just slightly different approach.