I have a few questions about dividing exponents. Can it be a fraction?
Like 18a^10b^13c^4 / 4a^7b^11c
and I got 9/2a^3b^2c^4.
For an equation that looks like this
3^7m^13n^6 / 3^5m^7n^5
would it be 1^2m^6n^1
Thanks for your help!
Do you perhaps mean "about simplifying expressions with exponents"...? Because I'm not seeing any division in the powers in what you've posted...?
Use the meaning of exponents to expand the expression, if you're not sure what to do:
. . . . .$\displaystyle \frac{18 a^10 b^13 c^4}{4 a^7 b^11 c}$
. . . . .$\displaystyle \left(\frac{18}{4}\right) \left(\frac{a^10}{a^7}\right) \left(\frac{b^13}{b^11}\right) \left(\frac{c^4}{c^1}\right)$
In the first fraction you can cancel the common factor of 2. In the second fraction, you have ten copies of $\displaystyle a$ on top, and seven underneath. How many extra do you have? (10 - 7 = 3) Where are they? (on top)
. . . . .$\displaystyle \left(\frac{9}{2}\right) \left(\frac{a^3}{1}\right) \left(\frac{b^13}{b^11}\right) \left(\frac{c^4}{c^1}\right)$
In the next fraction, you have thirteen copies of $\displaystyle b$ on top, and eleven underneath. How many extra do you have? (13 - 11 = 2) Where are they? (on top)
. . . . .$\displaystyle \left(\frac{9}{2}\right) \left(\frac{a^3}{1}\right) \left(\frac{b^2}{1}\right) \left(\frac{c^4}{c^1}\right)$
Simplify the last bit in the same way, and then put the fraction back together.
Note: Yes, you can end up with something more interesting in your denominator than just "1". It just so happens that this does not occur in the two examples you posted. But you should expect to see this in other exercises.
. . . . .$\displaystyle \frac{4^3 x^2 y}{4^5 x y^3}$
. . . . .$\displaystyle \left(\frac{4^3}{4^5}\right) \left(\frac{x^2}{x^1}\right) \left(\frac{y^1}{y^3}\right)$
. . . . .$\displaystyle \left(\frac{1}{4^2}\right) \left(\frac{x^1}{1}\right) \left(\frac{1}{y^2}\right)$
. . . . .$\displaystyle \frac{x}{4^2 y^2}\, =\, \frac{x}{16y^2}$
I'm also not sure about what you're asking, but I noticed you asked about exponents as fractions? The answer is yes, exponents can be fractions (technically, they always are fractions because anything can be expressed as a value over 1, because dividing by 1 does not change anything.)
But something like this is an example of how fractional exponents work:
4^(1/2)
sqrt(4)
= 2
Going to the power of 1/2 is the same as square rooting! This is an interesting application of fractional exponents.