Select the correct answer.

Rewrite the expression using only positive integer exponents.

[tex]\[ \left(m^{\frac{2}{3}} n^{-\frac{1}{3}}\right)^6 \][/tex]

A. [tex]\(\frac{n^2}{m^4}\)[/tex]

B. [tex]\(\frac{m^4}{n^2}\)[/tex]

C. [tex]\(\frac{n^{18}}{m^9}\)[/tex]

D. [tex]\(\frac{m^9}{n^{18}}\)[/tex]



Answer :

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]