Answer :
To factor the expression [tex]\(9x^4 - 64y^2\)[/tex], we can use the concept of the difference of squares.
### Step-by-Step Solution
1. Recognize the Difference of Squares:
The expression [tex]\(9x^4 - 64y^2\)[/tex] is a difference of squares. It takes the form [tex]\(a^2 - b^2\)[/tex], which can be factored using the formula:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
2. Identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
- [tex]\(9x^4\)[/tex] can be rewritten as [tex]\((3x^2)^2\)[/tex].
- [tex]\(64y^2\)[/tex] can be rewritten as [tex]\((8y)^2\)[/tex].
Therefore, you have:
[tex]\[
a = 3x^2 \quad \text{and} \quad b = 8y
\][/tex]
3. Apply the Difference of Squares Formula:
Using the identified values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], substitute them into the formula:
[tex]\[
9x^4 - 64y^2 = (3x^2 + 8y)(3x^2 - 8y)
\][/tex]
So, the factored form of the expression [tex]\(9x^4 - 64y^2\)[/tex] is:
[tex]\[
(3x^2 + 8y)(3x^2 - 8y)
\][/tex]
### Step-by-Step Solution
1. Recognize the Difference of Squares:
The expression [tex]\(9x^4 - 64y^2\)[/tex] is a difference of squares. It takes the form [tex]\(a^2 - b^2\)[/tex], which can be factored using the formula:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
2. Identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
- [tex]\(9x^4\)[/tex] can be rewritten as [tex]\((3x^2)^2\)[/tex].
- [tex]\(64y^2\)[/tex] can be rewritten as [tex]\((8y)^2\)[/tex].
Therefore, you have:
[tex]\[
a = 3x^2 \quad \text{and} \quad b = 8y
\][/tex]
3. Apply the Difference of Squares Formula:
Using the identified values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], substitute them into the formula:
[tex]\[
9x^4 - 64y^2 = (3x^2 + 8y)(3x^2 - 8y)
\][/tex]
So, the factored form of the expression [tex]\(9x^4 - 64y^2\)[/tex] is:
[tex]\[
(3x^2 + 8y)(3x^2 - 8y)
\][/tex]
