Let's simplify the given expression:
[tex]\[ \frac{28 p^9 q^{-5}}{12 p^{-6} q^7} \][/tex]
First, simplify the coefficient:
[tex]\[ \frac{28}{12} = \frac{28 \div 4}{12 \div 4} = \frac{7}{3} \][/tex]
Next, let's simplify the exponents of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] separately, using exponent rules.
### For [tex]\(p\)[/tex]:
The exponents of [tex]\(p\)[/tex] in the numerator and denominator are [tex]\(9\)[/tex] and [tex]\(-6\)[/tex] respectively.
[tex]\[ p^{9} \div p^{-6} = p^{9 - (-6)} = p^{9 + 6} = p^{15} \][/tex]
### For [tex]\(q\)[/tex]:
The exponents of [tex]\(q\)[/tex] in the numerator and denominator are [tex]\(-5\)[/tex] and [tex]\(7\)[/tex] respectively.
[tex]\[ q^{-5} \div q^{7} = q^{-5 - 7} = q^{-12} \][/tex]
Combining these results, the simplified expression is:
[tex]\[ \frac{7}{3} \times p^{15} \times q^{-12} \][/tex]
Thus, the expression is equivalent to:
[tex]\[ \frac{7 p^{15}}{3 q^{12}} \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{\frac{7 p^{15}}{3 q^{12}}} \][/tex]
