find the least common multiple: 3x^2, 12y, and 10x^3y^3. and please explain it to me.

Question

whpalmer4 · Answer

To find the LCM, whether of numbers or symbolic expressions like $10x^3y^3$, first break each item down into its prime factors.  Then construct the LCM by multiplying together the highest power of each different factor found.  I'll do an example:

Find the LCM of $2x^2, 3y^3, 4xyz$:
$$2x^2 = 2^1 * x^2$$$$3y^3=3*y^3$$$$4xyz=2^2*x^1*y^1*z^1$$

Different factors we found are $2,3,x,y,z$ with varying exponents.  Multiplying them all together with the highest exponents we found, we get:
$$2^2*3*x^2*y^3*z^1 = 12x^2y^3z$$

Any questions?

anonymous · Answer

so I just basically multiply the numbers together to get my answers

whpalmer4 · Answer

well, no, you have to multiply the right numbers and variables together to get your answer :-)

it's just like finding the LCM of 2, 4 and 8 — if you multiply them all together, you get a number which is (of course) a common multiple, but it isn't the LEAST common multiple...the LCM of 2, 4 and 8 is simply 8.

$$2=2^1$$$$4=2^2$$\[8=2^3]LCM = the product of the different factors taken with their highest exponents.  We only found $2$ as a factor, and the highest exponent of $2$ was $2^3$, so the LCM is $2^3 = 8$

2 4 6 8
4 8
8

as you can see, 8 is the first value that appears in each list of multiples.  Does that make sense?

whpalmer4 · Answer

If you like, you can look at the problem of doing LCM of your problem as first finding the LCM of the numbers, and multiplying that by the LCM of the variables.

anonymous · Answer

so I would do it like this.
3x^2 =  1 * 3 * x * x
12y = 2 * 2*3*y
10x^3y^3 =  2* 2 *5 * x * x *x *y *y*y

2*2*3*5*x*x*x*y*y*y = 60x^3y^3

whpalmer4 · Answer

Yes!  That's correct.

whpalmer4 · Answer

I would write it like this, just to make picking out the right exponents a bit easier:
$$3x^2 = 3^1*x^2$$$$12y=2^2*3^1*y^1$$$$10x^3y^3=2^1*5^1*x^3*y^3$$$$LCM = 2^2*3^1*5^1*x^3*y^3 = 4*3*5*x^3*y^3=60x^3y^3$$

whpalmer4 · Answer

but if you reliably get the right answer writing out long strings of repeated letters, that's what matters.