To simplify the expression [tex]\(\frac{m + (mn^2)^{1/3} + (m^2n)^{1/3}}{m - n} \times \left(1 - \frac{n^{1/3}}{m^{1/3}}\right)\)[/tex], let's analyze each part step-by-step:
1. Original Expression:
[tex]\[
\frac{m + (mn^2)^{1/3} + (m^2n)^{1/3}}{m - n} \times \left(1 - \frac{n^{1/3}}{m^{1/3}}\right)
\][/tex]
2. First Part: [tex]\( \frac{m + (mn^2)^{1/3} + (m^2n)^{1/3}}{m - n} \)[/tex]
Let's consider the numerator:
[tex]\[
m + (mn^2)^{1/3} + (m^2n)^{1/3}
\][/tex]
We can rewrite the cubic roots in their simplified forms:
[tex]\[
(mn^2)^{1/3} = m^{1/3} n^{2/3}, \quad (m^2n)^{1/3} = m^{2/3} n^{1/3}
\][/tex]
So, the numerator becomes:
[tex]\[
m + m^{1/3} n^{2/3} + m^{2/3} n^{1/3}
\][/tex]
3. Second Part: [tex]\( 1 - \frac{n^{1/3}}{m^{1/3}} \)[/tex]
We can leave this part as it is for now.
4. Combining the Parts:
Now, let's multiply the simplified numerator by the simplified second part:
[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]
5. Simplify the Combined Expression:
When we combine these parts, notice that the numerator can be reorganized:
[tex]\[
(m^{1/3} - n^{1/3}) \times (m + m^{1/3} n^{2/3} + m^{2/3} n^{1/3})
\][/tex]
Therefore, the entire expression can be written as:
[tex]\[
\frac{(m^{1/3} - n^{1/3})(m + m^{1/3} n^{2/3} + m^{2/3} n^{1/3})}{m^{1/3}(m - n)}
\][/tex]
6. Final Simplified Form:
Now, let's combine everything and simplify the expression:
[tex]\[
\frac{(m^{1/3} - n^{1/3})(m + m^{1/3} n^{2/3} + m^{2/3} n^{1/3})}{m^{1/3}(m - n)}
\][/tex]
Which can be further rewritten as:
[tex]\[
\left(m^{1/3} - n^{1/3}\right) \times \frac{m + (mn^2)^{1/3} + (m^2n)^{1/3}}{m^{1/3}(m - n)}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{(m^{1/3} - n^{1/3})(m + (mn^2)^{1/3} + (m^2n)^{1/3})}{m^{1/3}(m - n)}}
\][/tex]
