question about simplifying by dividing

[kid_A]

i've been trying the same two problems over and over again and i can't seem to get them right!!!!!

the problem that's bugging me the most is:

(18x) (6y^2) (4z)
______________
(18x) (2y)

for some reason i'm getting confused about how you go about simplifying this. even though i try to go through the steps the way the study guide i have (which contains this problem) explains them, i keep getting the wrong answer and i'm just stuck on how to simplify it.
can anyone help?!?
thanks!

-A

 

[royhaas]

If you show your work, we can tell where you are having trouble.

 

[Loren]

The problem as presented is the same as...

\(\displaystyle \frac{18x}{18x}\cdot \frac{6y^2}{2y}\cdot \frac{4z}{1}\)

Can you simplify that?

Error corrected.

 

[kid_A]

ok. i see what i was doing! thank you. i was basically trying to divide each part of the top equation by the entirety of the bottom equation... stupid me.
so say for example i have this:

20a+14b
________
2b

the answer would be 10a+7
______
b

and the b in the denominator stays because there was two and only one on top to be canceled out?

thank you again!

-A

 

[mmm4444bot]

COMMON ERROR ...

... so say for example i have this:

20a+14b
________
2b

the answer would be:

10a+7
______ ? Not correct.
b

... b in the denominator stays because there [is] two and only one on top to be canceled ... This reasoning is wrong.

Hi Kid:

Your reasoning on reducing algebraic fractions (i.e., rational expressions) is a common mistake.

\(\displaystyle \frac{a + b}{b}\)

You MAY NOT cancel the bs just because you "see" one on the top and bottom. That's against the rules of algebra.

You may only cancel FACTORS. If you want to cancel a symbol in the denominator (or numerator), then you must FIRST factor that same symbol out of the numerator (or denominator). If a symbol cannot be factored out, then it cannot be canceled.

\(\displaystyle \frac{20a + 14b}{2b}\)

\(\displaystyle \frac{2 \cdot (10a + 7b)}{2b}\)

\(\displaystyle \frac{10a + 7b}{b}\)

If you would like to cancel the b in the SINGLE term 7b, then THAT is acceptable. You would need to decompose the rational expression into a sum of two terms.

\(\displaystyle \frac{10a}{b} \;+\; 7\)

Cheers,

~ Mark