Question

Find the quotient.

-10d^4e^3f ^5 ÷ 15d 2^e ^2f

Answer 1

You wanna factor things out a bit: \[
-10d^4e^3f^5 = (-1)(2)(5)(d)(d)(d)(d)(e)(e)(e)(f)(f)(f)(f)(f) \\
15de^2f = (3)(5)(d)(e)(e)(f)
\]So our fraction is now: \[
\frac{(-1)(2)(5)(d)(d)(d)(d)(e)(e)(e)(f)(f)(f)(f)(f)} {(3)(5)(d)(e)(e)(f)}
\]
Then you wanna cancel things out.

Both have 5, cancel them: \[

\frac{ (-1)(2)(d)(d)(d)(d)(e)(e)(e)(f)(f)(f)(f)(f) }{ (3)(d)(e)(e)(f) }
\]
Both have d: \[

\frac{ (-1)(2)(d)(d)(d)(e)(e)(e)(f)(f)(f)(f)(f) }{ (3)(e)(e)(f) }
\]
Both have 2 'e's: \[

\frac{ (-1)(2)(d)(d)(d)(e)(f)(f)(f)(f)(f) }{ (3)(f) }
\]
Both have f: \[

\frac{ (-1)(2)(d)(d)(d)(e)(f)(f)(f)(f) }{ (3) }
\]
We simplify: \[
\frac{ -2d^3ef^4 }{ 3}
\]

Answer 2

You can do this even faster if you just subtract exponents on the variables.

Realize that \[
\frac{d^m}{d^n} = d^{m-n}
\]