Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Simplify: [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].

Sagot :

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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.