Questions: Exponents Power and quotient rules with positive exponents Simplify. [ left(frac2 m^7 n^54 n^2right)^3 ] Write your answer using only positive exponents.

Solution Steps

Start by simplifying the expression inside the parentheses using the quotient rule for exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\).

\[ \frac{2 m^{7} n^{5}}{4 n^{2}} = \frac{2}{4} \cdot m^{7} \cdot n^{5-2} \]

This simplifies to:

\[ \frac{1}{2} \cdot m^{7} \cdot n^{3} \]

Now apply the power rule, which states that \((a^m)^n = a^{m \cdot n}\), to the entire expression:

\[ \left(\frac{1}{2} \cdot m^{7} \cdot n^{3}\right)^{3} \]

This becomes:

\[ \left(\frac{1}{2}\right)^{3} \cdot (m^{7})^{3} \cdot (n^{3})^{3} \]

Simplify each term separately:

\[ \left(\frac{1}{2}\right)^{3} = \frac{1}{8} \]

\[ (m^{7})^{3} = m^{21} \]

\[ (n^{3})^{3} = n^{9} \]

Combine these results:

\[ \frac{1}{8} \cdot m^{21} \cdot n^{9} \]

Final Answer