Exponential growth and decay
Subtracting Rational Expressions Subtracting Rational Expressions
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- We're asked to subtract and state the difference
- in simplest form.
- So we're subtracting one rational expression from another.
- And just like the case when you are subtracting fractions,
- or adding fractions for that matter,
- you have to make sure that you have a common denomintator.
- And over here it looks like we have kind of the same components,
- but they're not quite the same thing.
- So we need to find the least common multiple of 7m^2n and 3mn^4,
- and then that we can use as a common denominator.
- Let's see if we can do that.
- So, let me find a common denominator:
- we have a negative sign out here and a common denominator over here.
- So a common denominator is going to have ...
- let's just look at each of the numbers.
- We have a 7 here and a 3 over here.
- So the common denominator has to be divisible by both 7 and 3.
- Or another way to think about it,
- the least common multiple of 7 and 3 is 21.
- So we're going to have to have 21 in our common denominator.
- And I'm getting that by encorporating the 7 and the 3.
- That's what gives me the 21.
- Now let's look at the m's.
- This denominator has 2 "m"s if you were to factor it.
- It's m times m.
- This only has one m.
- So as long as we're divisible by m^2
- then we're also going to be divisible by m
- So we just have to put an m^2 over here
- and an m^2 over here.
- Notice this is ... right now this is both divisible by
- 7, 3, m and m^2.
- So we're cool through the "m"s.
- Now let's look at the "n" terms.
- It has to be divisible by n,
- and it also has to be divisible by n^4.
- Well, if we're divisible by n^4,
- we're also divisible by n, since this is one of the factors of n^4
- This to the fourth power is n to the fourth.
- Anyway, we have to be divisible by n^4
- If we're divisible by n^4 we are divisible by n
- So that's our common denominator.
- We're divisible by 7m^2n, definitely divisible by 3mn^4.
- So that's our common denominator.
- So to go from 7m^2n, to this right over here,
- we have to multiply it by 3 times
- let's see, the m^2, we didn't have to multiply it by anything
- to get from n to n^4, we have to multiply it by n^3
- So we have to multiply this by 3n^3
- So we multiply the denominator by 3n^3 to get this over here
- we also have to multiply the numerator by 3n^3
- 3n^3 times 6m is 18m,
- one right over there
- and then we have an n^3
- now let's do it with the second term right over here
- or the thing we are subtracting from this first rational expression
- so to get from 3m,
- so let's see, to get from that to that
- what did we have to do?
- to get from a 3 to a 21
- you have to multiply it by a 7
- you have to multiply it by 7
- and to get from an m to an m^2 you have to multiply it by m
- And to get from n^4 to n^4, you didn't have to multiply by anything
- So to get from this expression in the denominator
- to this expression in the denominator
- you have to multiply it by 7m.
- So if you multiply the denominator by 7m
- you also have to ultiply the numerator by 7m
- And so you have 5 time 7 is 35
- and you have 1m, right over here,
- and then we have n^3
- and now we are ready to subtract
- since we have the same denominator
- Let me write the denominator here
- So the denominator is 21m^2n^4
- And in our numerator we have
- 18mn^3 minus 35mn^3
- Now this is interesting,
- because we have 18 of something
- of mn^3
- and we're going to subtract 35 of that same something
- mn^3
- we have the same power on m,
- same power on n
- So this is 18 of something -35 of that something
- which is going to be equal to what?
- 18 minus 35,
- this is just practicing subtracting negative numbers
- is -17
- Let me do this in a new color
- So this is going to be -17
- -17mn^3
- right, 18-35: 35-18 would be 17
- So if you swap them around you get -17
- yup that works
- and then all of that over 21m^2n^4.
- Now let's see if we can simplify this further
- both the numerator ...
- 17 is prime, so you can't simplify that with 21
- both the numerator and denominator are divisible by m
- So let's divide them both by m
- this becomes a 1, this becomes just an m
- both the numerator and denominator are divisible by n^3
- so you divide the numerator by n^3, that becomes 1
- you divide the denominator by n^3, this becomes an n
- We are left with, in simplest form,
- -17 over, I'll do it in the same colors, 21mn.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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