Question

Use the laws of exponents to simplify. Do not use negative exponents in your answer.
left parenthesis m Superscript five halves Baseline n Superscript negative four thirds Baseline right parenthesis Superscript four seventhsm
5
2n−
4
3
4
7
.
.
.
Question content area right
Part 1
left parenthesis m Superscript five halves Baseline n Superscript negative four thirds Baseline right parenthesis Superscript four seventhsm
5
2n−
4
3
4
7equals= enter your response here
​(Simplify your answer. Type exponential notation with positive rational​ exponents.)

Answer 1

To simplify \(\left( m^{\frac{5}{2}} n^{-\frac{4}{3}} \right)^{\frac{4}{7}}\), we will use the laws of exponents. Let's go through the steps:

1. **Apply the power of a power rule**: When raising a power to a power, you multiply the exponents.

\[
\left( m^{\frac{5}{2}} n^{-\frac{4}{3}} \right)^{\frac{4}{7}} = m^{\frac{5}{2} \cdot \frac{4}{7}} n^{-\frac{4}{3} \cdot \frac{4}{7}}
\]

2. **Calculate the exponents**:

- For \(m\):

\[
\frac{5}{2} \cdot \frac{4}{7} = \frac{5 \cdot 4}{2 \cdot 7} = \frac{20}{14} = \frac{10}{7}
\]

- For \(n\):

\[
-\frac{4}{3} \cdot \frac{4}{7} = -\frac{4 \cdot 4}{3 \cdot 7} = -\frac{16}{21}
\]

3. **Combine the results**:

\[
\left( m^{\frac{5}{2}} n^{-\frac{4}{3}} \right)^{\frac{4}{7}} = m^{\frac{10}{7}} n^{-\frac{16}{21}}
\]

4. **Since we do not want negative exponents**, we rewrite \(n^{-\frac{16}{21}}\) as \(\frac{1}{n^{\frac{16}{21}}}\):

\[
m^{\frac{10}{7}} n^{-\frac{16}{21}} = \frac{m^{\frac{10}{7}}}{n^{\frac{16}{21}}}
\]

Thus, the simplified expression is:

\[
\frac{m^{\frac{10}{7}}}{n^{\frac{16}{21}}}
\]

So the final answer is:

\[
\frac{m^{\frac{10}{7}}}{n^{\frac{16}{21}}}
\]