Alright, let's simplify the given expression step-by-step:
The expression we are working with is:
[tex]\[
\frac{m + \left(m n^2\right)^{1/3} + \left(m^2 n\right)^{1/3}}{m - n} \times \left(1 - \frac{n^{1/3}}{m^{1/3}}\right)
\][/tex]
First, let's rewrite each term to make it easier to manipulate. Let:
[tex]\[
A = m
\][/tex]
[tex]\[
B = \left(m n^2\right)^{1/3} = m^{1/3} n^{2/3}
\][/tex]
[tex]\[
C = \left(m^2 n\right)^{1/3} = m^{2/3} n^{1/3}
\][/tex]
Now the expression becomes:
[tex]\[
\frac{A + B + C}{m - n} \times \left(1 - \frac{n^{1/3}}{m^{1/3}}\right)
\][/tex]
Substitute [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
[tex]\[
\frac{m + m^{1/3} n^{2/3} + m^{2/3} n^{1/3}}{m - n} \times \left(1 - \frac{n^{1/3}}{m^{1/3}}\right)
\][/tex]
Next, observe the two parts of the expression that need to be simplified:
1. The fraction part:
[tex]\[
\frac{m + m^{1/3} n^{2/3} + m^{2/3} n^{1/3}}{m - n}
\][/tex]
2. The multiplicative factor:
[tex]\[
1 - \frac{n^{1/3}}{m^{1/3}}
\][/tex]
Combine these parts together:
[tex]\[
\left(\frac{m + m^{1/3} n^{2/3} + m^{2/3} n^{1/3}}{m - n}\right) \times \left(1 - \frac{n^{1/3}}{m^{1/3}}\right)
\][/tex]
We multiply them carefully:
[tex]\[
\left(\frac{m + m^{1/3} n^{2/3} + m^{2/3} n^{1/3}}{m - n}\right) \times \left(1 - \frac{n^{1/3}}{m^{1/3}}\right)
\][/tex]
The simplified form of this expression is:
[tex]\[
\boxed{\frac{(m^{1/3} - n^{1/3})(m + (m n^2)^{1/3} + (m^2 n)^{1/3})}{m^{1/3}(m - n)}}
\][/tex]
This completes our step-by-step solution.
