Question

Factor out the greatest common factor from the expression 9a²bᶾ – 12a²b⁴ – 24aᶾb

Answer 1

what is the greatest common factor for 9 12 and 24?

Answer 2

3

Answer 3

and what about a^2, a^2 and a^3

Answer 4

1

Answer 5

hmmm...is a^2 divisible by a?

Answer 6

no

Answer 7

carefully...a^2 is a times a, so the resulting number, whatever it is, should be divisible by a, right?

Answer 8

yes

Answer 9

so a^2 is divisible by a, but a^3 is too. now, are they both divisible by a^2?

Answer 10

no

Answer 11

why not? first you have a^2. it is definitely divisible by itself. then you have a^3 which is:
a^3 = a * a * a, and it should be divisible by a^2 since a^2 = a * a.
you may take different approach for a^3:
a^3 = a*a*a = a^2 * a, and this is obviously divisible by a^2, because we see that a^2 multiplied by a produces a^3, so a^3 must be divisible by a^2

Answer 12

ok

Answer 13

a^2, a^2 and a^3 cannot be divided with a^3, since a^2 isnt divisible with a^3 (try to explain why?)
use the same procedure for Bs

Answer 14

for example: 4^2 is 16 and 4^3 is 64, so there is no way 16 is divisible with 64. that's part of explanation. but there might be more explanations ...
now try to find the greatest divisor of b^3, b^4 and b

Answer 15

4^3 is 64 4^4 is 256

Answer 16

hmmm, i used number 4 just for and example. we need to find out if there is something that can divide b^3, b^4 and b, no matter what b is. do you have an idea?

Answer 17

ok, i'll try to explain:
we have 9a²bᶾ – 12a²b⁴ – 24aᶾb.
since we do not know a and b, we cannot mix them, they are like apples and pears. so we have to treat coefficients (numbers 9, 12, 24) separately from letters (a and b), and also, we need to treat a and b separately.
so first we see what is greatest common factor for 9 12 and 24, and we see it is 3
now we go for letter a. we have a^2, a^2 and a^3, in other words, we have:
a*a, a*a, a*a*a, so the greatest common divisor is a*a, that is a^2, since all of them are divisible with a*a.
Now for letter b. we have b^3, b^4 and b, in other words:
b*b*b, b*b*b*b, b, and greatest divisor for those is just b.
so, combining all their common things, we get that greatest common divisor is:
3 * a^2 * b
Is is now easier a bit?

Answer 18

a little

Answer 19

Sorry if i wasted your time...
We know that a^4 = a * a * a * a, just like 4a = a + a + a + a.

When you have a number, it is good to know what numbers need to be multiplied to get it, e.g.
24 = 2 * 2 * 2 * 3, 42 = 2 * 3 * 7.
This way, you can instantly see that greatest common divisor is made up from 2 and 3, that is
2*3, that is 6.

In the same way, we have (for our numbers):
9 a^2 b^4 = 3*3*a*a*b*b*b
12 a^2 b^4 = 2*2*3*a*a*b*b*b*b
24 a^3 b = 2*2*2*3*a*a*a*b
we see that all numbers contain: one 3, two a, one b. Greatest common divisor is greatest number that can divide them all, so it has to include all of these: 3, a, a, b, to be as big as possible. Here we have: Greatest common divisor = 3 * a * a * b = 3 * a^2 * b

Answer 20

no waste of time. my resposes are a little slow bc i have a 3 year old son who keeps calling for me. I appreciate all the help