Let's break down the given expression step by step and simplify it to find the equivalent expression using only positive integer exponents.
The given expression is:
[tex]\[
\left(m^{\frac{2}{3}} n^{-\frac{1}{3}}\right)^6
\][/tex]
1. Apply the power to each term inside the parenthesis:
We need to raise both the terms, [tex]\(m^\frac{2}{3}\)[/tex] and [tex]\(n^{-\frac{1}{3}}\)[/tex], to the power of 6.
2. Simplify each term:
For [tex]\(m\)[/tex]:
[tex]\[
\left(m^{\frac{2}{3}}\right)^6 = m^{\frac{2}{3} \cdot 6} = m^{4}
\][/tex]
For [tex]\(n\)[/tex]:
[tex]\[
\left(n^{-\frac{1}{3}}\right)^6 = n^{-\frac{1}{3} \cdot 6} = n^{-2}
\][/tex]
3. Combine the simplified terms:
Combining the results from the two terms, we get:
[tex]\[
m^{4} \cdot n^{-2}
\][/tex]
4. Rewrite with positive exponents:
To rewrite the expression with only positive exponents, we can use the property that [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex]:
[tex]\[
m^{4} \cdot n^{-2} = m^{4} \cdot \frac{1}{n^{2}} = \frac{m^{4}}{n^{2}}
\][/tex]
Thus, the given expression simplifies to:
[tex]\[
\frac{m^{4}}{n^{2}}
\][/tex]
So, the correct answer is:
B. [tex]\(\frac{m^4}{n^2}\)[/tex]
