To solve the given expression:

[ 7 \sqrt[5]{4x^3y} - 3 \sqrt[5]{32x^4y}, ]

we need to simplify each term and check for common radical components to combine them.

Step 1: Simplify each radical term

1. First term: $7 \sqrt[5]{4x^3y}$: This is already expressed in simplified form.

2. Second term: $3 \sqrt[5]{32x^4y}$: Notice that $32 = 2^5$. Breaking it down: [ 3 \sqrt[5]{32x^4y} = 3 \cdot 2 \cdot \sqrt[5]{x^4y} = 6 \sqrt[5]{x^4y}. ]

Step 2: Rewrite both terms with the same radical components

• The first term remains $7 \sqrt[5]{4x^3y}$.
• For consistency, factorize $4x^3y$ and $x^4y$ into smaller components:
  • $4x^3y = 2^2 \cdot x^3 \cdot y$,
  • $x^4y = x^4 \cdot y$.

Thus, the expression becomes: [ 7 \sqrt[5]{4x^3y} - 6 \sqrt[5]{x^4y}. ]

Step 3: Check for combining terms

The terms cannot be combined because the radicands ($4x^3y$ and $x^4y$) are not identical. Thus, the simplified form is:

[ 7 \sqrt[5]{4x^3y} - 6 \sqrt[5]{x^4y}. ]

Final Answer:

[ 7 \sqrt[5]{4x^3y} - 6 \sqrt[5]{x^4y}. ]

Let me know if you'd like detailed verification or have questions!

Related Questions:

1. How do you factorize radicands with multiple variables for simplification?
2. What are the rules for combining terms involving radicals?
3. How do fractional exponents relate to radicals?
4. Can you simplify $\sqrt[5]{16x^5y}$?
5. What happens when the radicands are identical in radical expressions?

Tip:

Always check if the radicands (inside the radicals) can be factored or simplified further to make combining terms easier!