To simplify the expression [tex]\(\left(m^{\frac{2}{3}} n^{-\frac{1}{3}}\right)^6\)[/tex] and rewrite it using only positive integer exponents, we'll follow these steps:
1. Distribute the exponent 6 to each term inside the parentheses:
The general rule for exponents when dealing with multiplication inside parentheses is [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]. Applying this rule, we get:
[tex]\[
\left(m^{\frac{2}{3}} n^{-\frac{1}{3}}\right)^6 = \left(m^{\frac{2}{3}}\right)^6 \cdot \left(n^{-\frac{1}{3}}\right)^6
\][/tex]
2. Multiply the exponents:
To simplify each term, we multiply the exponents. For [tex]\(m^{\frac{2}{3}}\)[/tex]:
[tex]\[
\left(m^{\frac{2}{3}}\right)^6 = m^{\frac{2}{3} \cdot 6} = m^{\frac{12}{3}} = m^4
\][/tex]
Similarly, for [tex]\(n^{-\frac{1}{3}}\)[/tex]:
[tex]\[
\left(n^{-\frac{1}{3}}\right)^6 = n^{-\frac{1}{3} \cdot 6} = n^{-\frac{6}{3}} = n^{-2}
\][/tex]
3. Rewrite the expression with positive exponents:
Recall that a negative exponent indicates the reciprocal of the base raised to the positive exponent:
[tex]\[
n^{-2} = \frac{1}{n^2}
\][/tex]
Combining the results, we get:
[tex]\[
m^4 \cdot n^{-2} = m^4 \cdot \frac{1}{n^2} = \frac{m^4}{n^2}
\][/tex]
Thus, the simplified expression with only positive integer exponents is:
[tex]\(\boxed{\frac{m^4}{n^2}}\)[/tex]
Therefore, the correct answer is:
D. [tex]\(\frac{m^4}{n^2}\)[/tex]
