Answer :
To factor the expression [tex]\(9x^4 - 64y^2\)[/tex], we can start by recognizing its resemblance to a difference of squares. The expression can be seen as a difference of squares because both [tex]\(9x^4\)[/tex] and [tex]\(64y^2\)[/tex] are perfect squares. Here's how we can factor it step by step:
1. Identify the squares:
- [tex]\(9x^4\)[/tex] is a perfect square. It can be written as [tex]\((3x^2)^2\)[/tex], because [tex]\(3x^2\)[/tex] squared equals [tex]\(9x^4\)[/tex].
- [tex]\(64y^2\)[/tex] is also a perfect square. It can be written as [tex]\((8y)^2\)[/tex], because [tex]\(8y\)[/tex] squared equals [tex]\(64y^2\)[/tex].
2. Apply the difference of squares formula:
The difference of squares formula is given by:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Using the values identified:
- Let [tex]\(a = 3x^2\)[/tex]
- Let [tex]\(b = 8y\)[/tex]
Substitute into the formula:
[tex]\[
(3x^2)^2 - (8y)^2 = (3x^2 - 8y)(3x^2 + 8y)
\][/tex]
Therefore, the factored form of [tex]\(9x^4 - 64y^2\)[/tex] is [tex]\((3x^2 - 8y)(3x^2 + 8y)\)[/tex].
1. Identify the squares:
- [tex]\(9x^4\)[/tex] is a perfect square. It can be written as [tex]\((3x^2)^2\)[/tex], because [tex]\(3x^2\)[/tex] squared equals [tex]\(9x^4\)[/tex].
- [tex]\(64y^2\)[/tex] is also a perfect square. It can be written as [tex]\((8y)^2\)[/tex], because [tex]\(8y\)[/tex] squared equals [tex]\(64y^2\)[/tex].
2. Apply the difference of squares formula:
The difference of squares formula is given by:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Using the values identified:
- Let [tex]\(a = 3x^2\)[/tex]
- Let [tex]\(b = 8y\)[/tex]
Substitute into the formula:
[tex]\[
(3x^2)^2 - (8y)^2 = (3x^2 - 8y)(3x^2 + 8y)
\][/tex]
Therefore, the factored form of [tex]\(9x^4 - 64y^2\)[/tex] is [tex]\((3x^2 - 8y)(3x^2 + 8y)\)[/tex].
