Answer :
To factor the expression [tex]\(9x^4 - 64y^2\)[/tex], we can use the difference of squares method. The difference of squares is a special algebraic pattern where [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]. Here’s how we can apply this pattern to the given problem:
1. Identify the squares:
- The term [tex]\(9x^4\)[/tex] is a perfect square because it can be written as [tex]\((3x^2)^2\)[/tex].
- The term [tex]\(64y^2\)[/tex] is also a perfect square because it can be written as [tex]\((8y)^2\)[/tex].
2. Express the expression as a difference of squares:
[tex]\[
9x^4 - 64y^2 = (3x^2)^2 - (8y)^2
\][/tex]
3. Apply the difference of squares formula:
- According to the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], we identify [tex]\(a = 3x^2\)[/tex] and [tex]\(b = 8y\)[/tex].
4. Write the factored form:
- Using the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the formula, we get:
[tex]\[
(3x^2 - 8y)(3x^2 + 8y)
\][/tex]
So, the factored form of [tex]\(9x^4 - 64y^2\)[/tex] is [tex]\((3x^2 - 8y)(3x^2 + 8y)\)[/tex].
1. Identify the squares:
- The term [tex]\(9x^4\)[/tex] is a perfect square because it can be written as [tex]\((3x^2)^2\)[/tex].
- The term [tex]\(64y^2\)[/tex] is also a perfect square because it can be written as [tex]\((8y)^2\)[/tex].
2. Express the expression as a difference of squares:
[tex]\[
9x^4 - 64y^2 = (3x^2)^2 - (8y)^2
\][/tex]
3. Apply the difference of squares formula:
- According to the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], we identify [tex]\(a = 3x^2\)[/tex] and [tex]\(b = 8y\)[/tex].
4. Write the factored form:
- Using the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the formula, we get:
[tex]\[
(3x^2 - 8y)(3x^2 + 8y)
\][/tex]
So, the factored form of [tex]\(9x^4 - 64y^2\)[/tex] is [tex]\((3x^2 - 8y)(3x^2 + 8y)\)[/tex].
