To factor the expression \(54x^7 + 16xy^3\), we can follow these steps:

1. **Find the greatest common factor (GCF):**
   - The GCF of the numerical coefficients 54 and 16 is 2.
   - The variable part contains \(x\) in both terms, with the lowest power of \(x\) being \(x\). Thus, the GCF of the variable part is \(x\).

   Therefore, the GCF of the entire expression is \(2x\).

2. **Factor out the GCF:**
   - Now we divide each term by \(2x\):
     \[
     54x^7 + 16xy^3 = 2x(27x^6 + 8y^3)
     \]

So, the factored form of \(54x^7 + 16xy^3\) is:

\[
\boxed{2x(27x^6 + 8y^3)}
\]

Would you like more details or have any questions about this factorization?

Here are some related questions for further exploration:

1. What is the greatest common factor of \(54x^7\) and \(16xy^3\)?
2. How would you factor a cubic expression like \(x^3 + 8y^3\)?
3. What is the difference between factoring and expanding expressions?
4. How do you simplify algebraic expressions with multiple variables?
5. Can we factor \(27x^6 + 8y^3\) further if necessary?

**Tip:** Always check for a common factor in both the numerical and variable parts of each term when factoring expressions.

Learn how to factor the polynomial expression 54x^7 + 16xy^3 by identifying and factoring out the greatest common factor (GCF). This step-by-step solution demonstrates core algebraic concepts and is suitable for students in Grades 8-10.

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