Sure, let's simplify the given expression step by step:
Given expression:
[tex]\[ \frac{p^6 q^4}{p^3 q^{16}} \][/tex]
1. Simplify the [tex]\( p \)[/tex]-terms:
Apply the rule of exponents for division, which states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{p^6}{p^3} = p^{6-3} = p^3
\][/tex]
2. Simplify the [tex]\( q \)[/tex]-terms:
Similarly, apply the same rule of exponents for the [tex]\( q \)[/tex]-terms:
[tex]\[
\frac{q^4}{q^{16}} = q^{4-16} = q^{-12}
\][/tex]
3. Combine the simplified parts:
After simplifying both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] terms, the expression can be written as:
[tex]\[
p^3 q^{-12}
\][/tex]
4. Rewrite using positive exponents:
To rewrite using positive exponents, recall that [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]:
[tex]\[
p^3 q^{-12} = \frac{p^3}{q^{12}}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{\frac{p^3}{q^{12}}}
\][/tex]
So, the correct answer from the given choices is:
[tex]\[
\frac{p^3}{q^{12}}
\][/tex]
