Questions: Exponents Power and quotient rules with positive exponents Simplify. (m^5 n^7 / m n^2)^5 Write your answer using only positive exponents.

Solution Steps

To simplify the given expression using the power and quotient rules for exponents, follow these steps:

1. Simplify the fraction inside the parentheses by subtracting the exponents of like bases.
2. Apply the power rule by raising the simplified expression to the power outside the parentheses.

We start with the expression

\[ \left(\frac{m^{5} n^{7}}{m n^{2}}\right)^{5}. \]

Using the quotient rule for exponents, we simplify the fraction inside the parentheses:

\[ \frac{m^{5}}{m} = m^{5-1} = m^{4} \]

and

\[ \frac{n^{7}}{n^{2}} = n^{7-2} = n^{5}. \]

Thus, we can rewrite the expression as

\[ \left(m^{4} n^{5}\right)^{5}. \]

Next, we apply the power rule, which states that \((a^{m})^{n} = a^{m \cdot n}\). Therefore, we have:

\[ \left(m^{4} n^{5}\right)^{5} = m^{4 \cdot 5} n^{5 \cdot 5} = m^{20} n^{25}. \]

Final Answer