To find an expression equivalent to [tex]\(\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}\)[/tex], let's break down the problem into simpler steps:
1. Simplify the numerical coefficient:
[tex]\[
\frac{28}{12}
\][/tex]
Simplifying [tex]\(\frac{28}{12}\)[/tex] gives [tex]\(\frac{7}{3}\)[/tex].
2. Combine the exponents for [tex]\(p\)[/tex]:
[tex]\[
\frac{p^9}{p^{-6}}
\][/tex]
When dividing like bases, you subtract the exponents:
[tex]\[
p^{9 - (-6)} = p^{9 + 6} = p^{15}
\][/tex]
3. Combine the exponents for [tex]\(q\)[/tex]:
[tex]\[
\frac{q^{-5}}{q^7}
\][/tex]
Similarly, for [tex]\(q\)[/tex],
[tex]\[
q^{-5 - 7} = q^{-12}
\][/tex]
Now, putting all these simplified terms together, we get:
[tex]\[
\frac{28 p^9 q^{-5}}{12 p^{-6} q^7} = \frac{7}{3} p^{15} q^{-12}
\][/tex]
Thus, the equivalent expression is:
[tex]\[
\frac{7 p^{15} q^{-12}}{3}
\][/tex]
Looking at the options given:
1. [tex]\(\frac{2}{p^{15} q^{12}}\)[/tex]
2. [tex]\(\frac{7 p^{15}}{3 q^{12}}\)[/tex]
3. [tex]\(\frac{2 q^{12}}{p^{15}}\)[/tex]
4. [tex]\(\frac{7 p^{15} q^{12}}{3}\)[/tex]
The correct equivalent expression matches option 2:
[tex]\[
\frac{7 p^{15}}{3 q^{12}}
\][/tex]
