Sure, let's break down the given expression step-by-step and simplify it.
The expression we need to simplify 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]
### Step 1: Break Down the Terms
Let's rewrite the terms to make the exponents clearer:
1. [tex]\(\left(m n^2\right)^{1/3} = (m^{1/3} \times n^{2/3})\)[/tex]
2. [tex]\(\left(m^2 n\right)^{1/3} = (m^{2/3} \times n^{1/3})\)[/tex]
So we can rewrite the numerator as:
[tex]\[
m + m^{1/3} \times n^{2/3} + m^{2/3} \times n^{1/3}
\][/tex]
### Step 2: Simplify the Fraction
Now consider the fraction:
[tex]\[
\frac{m + m^{1/3} \times n^{2/3} + m^{2/3} \times n^{1/3}}{m - n}
\][/tex]
### Step 3: Factor and Simplify
We factor out the expression:
[tex]\[
\left(1 - \frac{n^{1/3}}{m^{1/3}}\right)
\][/tex]
Multiply the two simplified parts together:
### Step 4: Combine the Two Parts
We combine the two terms in the original expression:
[tex]\[
\left(1 - \frac{n^{1/3}}{m^{1/3}}\right)
\][/tex]
The complete expression becomes:
[tex]\[
\frac{m + m^{1/3} \times n^{2/3} + m^{2/3} \times n^{1/3}}{m - n}
\][/tex]
When we combine those parts, the simplified version of the whole expression is:
[tex]\[
\frac{(m^{1/3} - n^{1/3})(m + (mn^2)^{1/3} + (m^2n)^{1/3})}{m^{1/3}(m - n)}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
(m^{0.333333333333333} - n^{0.333333333333333}) \cdot \frac{m + (m \cdot n^2)^{0.333333333333333} + (m^2 \cdot n)^{0.333333333333333}}{m^{0.333333333333333} \cdot (m - n)}
\][/tex]
