Factor completely.
x Superscript 16 Baseline minus y Superscript 8
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Part 1
Select the correct choice below and fill in any answer boxes within your choice.
A.x Superscript 16 Baseline minus y Superscript 8 equals enter your response here
x Superscript 16 Baseline minus y Superscript 8equals
  
enter your response here 
B.
The polynomial is prime.

We are asked to factor the expression \( x^{16} - y^8 \). This is a difference of squares, which takes the general form:

\[
a^2 - b^2 = (a - b)(a + b)
\]

### Step 1: Express the given terms as squares:
\[
x^{16} = (x^8)^2 \quad \text{and} \quad y^8 = (y^4)^2
\]

Thus, we rewrite the expression as:
\[
(x^8)^2 - (y^4)^2
\]

### Step 2: Apply the difference of squares formula:
\[
(x^8 - y^4)(x^8 + y^4)
\]

### Step 3: Further factor \( x^8 - y^4 \):
We recognize that \( x^8 - y^4 \) is again a difference of squares:
\[
x^8 - y^4 = (x^4 - y^2)(x^4 + y^2)
\]

So now the expression becomes:
\[
(x^4 - y^2)(x^4 + y^2)(x^8 + y^4)
\]

### Step 4: Factor \( x^4 - y^2 \):
Again, \( x^4 - y^2 \) is a difference of squares:
\[
x^4 - y^2 = (x^2 - y)(x^2 + y)
\]

Now the complete factorization is:
\[
(x^2 - y)(x^2 + y)(x^4 + y^2)(x^8 + y^4)
\]

### Final Answer:
\[
x^{16} - y^8 = (x^2 - y)(x^2 + y)(x^4 + y^2)(x^8 + y^4)
\]

Would you like more details on any of the steps?

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Here are 5 related questions:
1. How do you factor expressions like \( x^4 - y^4 \)?
2. What is the difference of squares formula and when do you use it?
3. Can the sum of squares be factored over real numbers?
4. How do you recognize when to apply difference of squares?
5. How would the factorization change if we had \( x^{16} + y^8 \)?

**Tip:** Always check if you can apply the difference of squares multiple times for complete factorization!

Learn how to factor the expression x^16 - y^8 using the difference of squares formula. This detailed step-by-step solution covers multiple levels of factoring, from x^16 - y^8 down to simpler terms. Ideal for students in Grades 10-12 studying advanced algebra concepts.

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