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Answer :
To factor the expression [tex]\(9x^4 - 64y^2\)[/tex], we can use the difference of squares formula. The difference of squares states that for any two terms [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the expression [tex]\(a^2 - b^2\)[/tex] can be factored as [tex]\((a - b)(a + b)\)[/tex].
Here's a step-by-step breakdown:
1. Identify the terms as squares:
- [tex]\(9x^4\)[/tex] is a perfect square. It can be written as [tex]\((3x^2)^2\)[/tex].
- [tex]\(64y^2\)[/tex] is also a perfect square. It can be written as [tex]\((8y)^2\)[/tex].
2. Apply the difference of squares:
With the expression [tex]\(9x^4 - 64y^2\)[/tex] rearranged as [tex]\((3x^2)^2 - (8y)^2\)[/tex], we can apply the difference of squares formula:
[tex]\[
(a^2 - b^2) = (a - b)(a + b)
\][/tex]
For our expression, [tex]\(a = 3x^2\)[/tex] and [tex]\(b = 8y\)[/tex].
3. Write the factored form:
Using the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the expression factors to:
[tex]\[
(3x^2 - 8y)(3x^2 + 8y)
\][/tex]
Thus, the factored form of [tex]\(9x^4 - 64y^2\)[/tex] is [tex]\((3x^2 - 8y)(3x^2 + 8y)\)[/tex].
Here's a step-by-step breakdown:
1. Identify the terms as squares:
- [tex]\(9x^4\)[/tex] is a perfect square. It can be written as [tex]\((3x^2)^2\)[/tex].
- [tex]\(64y^2\)[/tex] is also a perfect square. It can be written as [tex]\((8y)^2\)[/tex].
2. Apply the difference of squares:
With the expression [tex]\(9x^4 - 64y^2\)[/tex] rearranged as [tex]\((3x^2)^2 - (8y)^2\)[/tex], we can apply the difference of squares formula:
[tex]\[
(a^2 - b^2) = (a - b)(a + b)
\][/tex]
For our expression, [tex]\(a = 3x^2\)[/tex] and [tex]\(b = 8y\)[/tex].
3. Write the factored form:
Using the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the expression factors to:
[tex]\[
(3x^2 - 8y)(3x^2 + 8y)
\][/tex]
Thus, the factored form of [tex]\(9x^4 - 64y^2\)[/tex] is [tex]\((3x^2 - 8y)(3x^2 + 8y)\)[/tex].
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