What is the factored form of `16x^(6)-8x^(3)y^(3)+y^(6)`?

Options

A. `(4x^(3)-y^(3))^(2)`
B. `(4x^(3)+y^(3))^(2)`
C. `(4x^(3)-y)^(6)`
D. `(16x^(2)+y)^(3)`

Explanation

To factor the expression \(16x^6 - 8x^3y^3 + y^6\), we can follow these steps:

### Step 1: Identify the terms
The expression consists of three terms:
- \(16x^6\)
- \(-8x^3y^3\)
- \(y^6\)

### Step 2: Recognize the structure
Notice that \(16x^6\) is a perfect square, as it can be expressed as \((4x^3)^2\), and \(y^6\) is also a perfect square, expressed as \((y^3)^2\). The middle term \(-8x^3y^3\) can be rewritten in terms of these squares.

### Step 3: Rewrite the middle term
The middle term \(-8x^3y^3\) can be rewritten as:
\[ -2 \cdot (4x^3) \cdot (y^3) \]
This matches the form of the expansion of a square of a binomial, which is \(a^2 - 2ab + b^2\).

### Step 4: Identify \(a\) and \(b\)
From the expression, we can identify:
- \(a = 4x^3\)
- \(b = y^3\)

### Step 5: Write the expression as a square
Using the identified \(a\) and \(b\), we can express the original expression as:
\[ (4x^3 - y^3)^2 \]

### Step 6: Final factored form
Thus, the factored form of the expression \(16x^6 - 8x^3y^3 + y^6\) is:
\[ (4x^3 - y^3)^2 \]

### Summary
The factored form of \(16x^6 - 8x^3y^3 + y^6\) is \((4x^3 - y^3)^2\).

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