To divide [tex]\( \frac{12x y^3 z^6}{4 x^5 y z^{12}} \)[/tex]:
1. Divide the coefficients [tex]\(12\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[
\frac{12}{4} = 3.
\][/tex]
2. Handle the variables. Use the rules of exponents for division: [tex]\(a^m / a^n = a^{m-n}\)[/tex].
- For [tex]\(x\)[/tex]:
[tex]\[
x^1 / x^5 = x^{1-5} = x^{-4}.
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
y^3 / y^1 = y^{3-1} = y^2.
\][/tex]
- For [tex]\(z\)[/tex]:
[tex]\[
z^6 / z^{12} = z^{6-12} = z^{-6}.
\][/tex]
3. Combine the results:
[tex]\[
\frac{12 x y^3 z^6}{4 x^5 y z^{12}} = 3 x^{-4} y^2 z^{-6}.
\][/tex]
So, the final result is:
[tex]\[
3 x^{-4} y^2 z^{-6}.
\][/tex]
In other words, when you divide [tex]\(12 x y^3 z^6\)[/tex] by [tex]\(4 x^5 y z^{12}\)[/tex], you get [tex]\(3 x^{-4} y^2 z^{-6}\)[/tex].
