ANSWER
[tex] \frac{ {m}^{2} {n}^{ - 5} }{ {m}^{ 7} {n}^{ - 17} } = \frac{ {n}^{12} }{ {m}^{5} } [/tex]
EXPLANATION
The given expression is
[tex] \frac{ {m}^{2} {n}^{ - 5} }{ {m}^{ 7} {n}^{ - 17} } [/tex]
Recall the following law of exponents,
[tex] \frac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n} [/tex]
We apply this law to obtain:
[tex] \frac{ {m}^{2} {n}^{ - 5} }{ {m}^{ 7} {n}^{ - 17} } = {m}^{2 - 7} {n}^{ - 5 - - 17} [/tex]
This simplifies to,
[tex] \frac{ {m}^{2} {n}^{ - 5} }{ {m}^{ 7} {n}^{ - 17} } = {m}^{ 2- 7} {n}^{ - 5 + 17} [/tex]
[tex] \frac{ {m}^{2} {n}^{ - 5} }{ {m}^{ 7} {n}^{ - 17} } = {m}^{ - 5} {n}^{ 12} [/tex]
Recall again that,
[tex] {a}^{ - m} = \frac{1}{ {a}^{m} } [/tex]
This implies that,
[tex] \frac{ {m}^{2} {n}^{ - 5} }{ {m}^{ 7} {n}^{ - 17} } = \frac{ {n}^{12} }{ {m}^{5} } [/tex]
The correct answer is D.
