Find the greatest common factor

of these monomials. And when they say monomials,

that’s just a fancy word for saying a one term expression. Each of these only, obviously,

have one term in them. Now to find the greatest common

factor of these, the way I think about it is, I like

to break up each of these terms into their constituent

parts. Make them a product of the

simplest things possible. For regular numbers like 10, to

me that means break them up into their prime factors, and

for these variable expressions like cd squared, break it up

into the product of the most simple variable, for example,

c times d times d. So let’s do that for each of

them and see what the greatest common factor is. Where do these overlap in

terms of their factor? And we care about the

greatest overlap. So let’s do this first one. 10 cd squared, what

is that equal to? Well 10 is equal to 2 times 5. You could do a factoring tree

here, but these are pretty straightforward numbers

to factor into. They’re prime factors. So 10 is 2 times 5, c, all you

can do is break that, you could just write that as

a c, you can’t really simplify that anymore. And d squared can be written

as d times d. So I have essentially broken

10cd squared into this, into the product of kind of the

smallest constituents that I could think of. The prime factors of 10,

and then c, and then d. Now let’s do 5cd. Well 5cd, 5 is prime, so its

prime factorization is literally just 5. c you can’t break that down

anymore, that’s just a c, and then times a d. So we really didn’t do anything

to this expression right there. And then finally you have 25c

to the third d squared. Well 25 is 5 times 5, and then

we have times c times c times c, that’s what c to the third

is, and then we have times d squared, times d times d. Now, what is the greatest common

factor, or what is the greatest common overlap between

these three things? Well they all have a 5. Let me circle them. You have a 5 there, you have a

5 there, you have a 5 there. They all have at least one c. You have one c there,

one c there, and then another c there. And they all have

at least one d. You have a d there, you have

a d there, and then you have a d there. Now they don’t all have a second

d, only the first one and the third one

have a second d. And they all don’t have a second

or third c, only this last one has a second

or third c. So we’re essentially done. The greatest common

factor is 5cd. In fact you can’t have a greater

number than 5cd be a common factor, because the

largest factor of 5cd is 5cd. So the greatest common factor

of these three monomials, or these three expressions,

is 5cd. The largest number of factors

that overlaps with all three of these expressions is

a 5, one c, and one d.

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thanks

Isn't there a easier way as distribution of the number you don't have to do all multiplying and dividing