Question

Factor the difference of two squares.
256 x Superscript 4 Baseline minus 81
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Part 1
256 x Superscript 4 Baseline minus 81equals
  
enter your response here  ​(Factor completely.)

Answer 1

To factor the expression \( 256x^4 - 81 \), we recognize it as a difference of two squares. The general form for the difference of two squares is:

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

In this case:

- \( a^2 = 256x^4 \) so \( a = 16x^2 \) (since \( 16^2 = 256 \) and \( (x^2)^2 = x^4 \))
- \( b^2 = 81 \) so \( b = 9 \) (since \( 9^2 = 81 \))

Now we can apply the difference of squares formula:

\[
256x^4 - 81 = (16x^2 - 9)(16x^2 + 9)
\]

Next, we can observe that \( 16x^2 - 9 \) is also a difference of squares:

\[
16x^2 - 9 = (4x - 3)(4x + 3)
\]

Thus, we can factor it completely.

Therefore, the fully factored form of \( 256x^4 - 81 \) is:

\[
(4x - 3)(4x + 3)(16x^2 + 9)
\]

So the final answer is:

\[
256x^4 - 81 = (4x - 3)(4x + 3)(16x^2 + 9)
\]