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To factor the expression \( x^{16} - y^{10} \) completely, let's observe that it is a difference of two terms with even exponents, which can be expressed as a difference of squares multiple times. Here’s the step-by-step factorization:

1. **Write as a difference of squares:**
   \[
   x^{16} - y^{10} = \left( x^8 \right)^2 - \left( y^5 \right)^2
   \]
   This can be factored using the difference of squares formula: \( a^2 - b^2 = (a + b)(a - b) \).

2. **Apply the difference of squares formula:**
   \[
   x^{16} - y^{10} = \left( x^8 + y^5 \right)(x^8 - y^5)
   \]
   
3. **Check for further factorization:**
   Neither \( x^8 + y^5 \) nor \( x^8 - y^5 \) can be factored further using real numbers, so this is the completely factored form over the real numbers.

### Final Answer:
\[
x^{16} - y^{10} = \left( x^8 + y^5 \right)(x^8 - y^5)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you factor a binomial if the exponents are odd?
2. Can the difference of two odd powers be factored?
3. What other methods exist for factoring expressions with high powers?
4. How does the difference of cubes formula differ from the difference of squares?
5. Can \( x^{16} - y^{10} \) be factored further over complex numbers?

### Tip:
Remember that the difference of squares formula can be applied repeatedly if both resulting terms can still be expressed as squares.

Factor the following binomial completely: x^16 - y^10

Learn how to factor the binomial x^16 - y^10 completely using the difference of squares formula. This detailed solution breaks down each step, demonstrating how to apply the difference of squares to high exponents. Suitable for high school students in Grades 10-12.

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