To simplify the given expression:
[tex]\[ \frac{m + \left(m n^2\right)^3 + \left(m^2 n\right)^3}{m-n} \times \left(1 - \frac{n^3}{m^3}\right) \][/tex]
we proceed step-by-step.
### Step 1: Simplify the numerator
The expression within the numerator is:
[tex]\[ m + (m n^2)^3 + (m^2 n)^3 \][/tex]
Rewrite the terms individually:
[tex]\[ (m n^2)^3 = m^3 n^6 \][/tex]
[tex]\[ (m^2 n)^3 = m^6 n^3 \][/tex]
Hence, the numerator becomes:
[tex]\[ m + m^3 n^6 + m^6 n^3 \][/tex]
### Step 2: Simplify the entire fraction
The fraction we are dealing with is:
[tex]\[ \frac{m + m^3 n^6 + m^6 n^3}{m - n} \][/tex]
### Step 3: Consider the second part of the expression separately
The second part of the expression is:
[tex]\[ 1 - \frac{n^3}{m^3} \][/tex]
This can be rewritten as:
[tex]\[ \frac{m^3 - n^3}{m^3} \][/tex]
### Step 4: Combine the two parts
Now, multiply the simplified fraction by the simplified second part:
[tex]\[ \left( \frac{m + m^3 n^6 + m^6 n^3}{m - n} \right) \times \left( \frac{m^3 - n^3}{m^3} \right) \][/tex]
### Step 5: Use polynomial identities
Recall that [tex]\( m^3 - n^3 \)[/tex] can be factored as:
[tex]\[ m^3 - n^3 = (m - n)(m^2 + mn + n^2) \][/tex]
So the expression becomes:
[tex]\[ \frac{m + m^3 n^6 + m^6 n^3}{m - n} \times \frac{(m - n)(m^2 + mn + n^2)}{m^3} \][/tex]
### Step 6: Cancel common terms
Observe that [tex]\( m - n \)[/tex] appears in both the numerator and the denominator and can be canceled out. This yields:
[tex]\[ \frac{m + m^3 n^6 + m^6 n^3}{1} \times \frac{m^2 + mn + n^2}{m^3} \][/tex]
### Step 7: Distribute and simplify the remaining expression
Let's multiply out the terms:
[tex]\[ \big(m + m^3 n^6 + m^6 n^3\big) \times \frac{(m^2 + mn + n^2)}{m^3} \][/tex]
Distribute each part of the first expression through the second fraction:
[tex]\[ \left(\frac{m (m^2 + mn + n^2)}{m^3}\right) + \left(\frac{m^3 n^6 (m^2 + mn + n^2)}{m^3}\right) + \left(\frac{m^6 n^3 (m^2 + mn + n^2)}{m^3}\right) \][/tex]
This simplifies to:
[tex]\[ \frac{m^3 + m^2 mn + mn^2}{m^3} + \frac{m^2 n^6 + mn^7 + n^8}{1} + \frac{m^5 n^3 + m^4 n^4 + m^3 n^5}{m} \][/tex]
### Step 8: Combine like terms
This will further simplify and combine like terms to eventually yield:
[tex]\[ \frac{(m^3 - n^3)(m^5 n^3 + m^2 n^6 + 1)}{m^2 (m - n)} \][/tex]
Thus, the final simplified form of the expression is:
[tex]\[ \boxed{\frac{(m^3 - n^3)(m^5 n^3 + m^2 n^6 + 1)}{m^2 (m - n)}} \][/tex]
