Answer :
Let's solve the problem of finding which expressions are equivalent to [tex]\(25x^4 - 64\)[/tex].
1. First Expression: [tex]\(25x^4 + 40x - 40x - 64\)[/tex]
To find if this expression is equivalent to [tex]\(25x^4 - 64\)[/tex], simplify it:
[tex]\[
25x^4 + 40x - 40x - 64 = 25x^4 + (40x - 40x) - 64 = 25x^4 - 64
\][/tex]
This expression is equivalent to [tex]\(25x^4 - 64\)[/tex].
2. Second Expression: [tex]\(25x^4 + 13x - 13x - 64\)[/tex]
Simplify this expression:
[tex]\[
25x^4 + 13x - 13x - 64 = 25x^4 + (13x - 13x) - 64 = 25x^4 - 64
\][/tex]
This expression is also equivalent to [tex]\(25x^4 - 64\)[/tex].
3. Third Expression: [tex]\((5x^2 + 8)(5x^2 - 8)\)[/tex]
This expression is in the form of a difference of squares, which states:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Here, [tex]\(a = 5x^2\)[/tex] and [tex]\(b = 8\)[/tex], therefore:
[tex]\[
(5x^2 + 8)(5x^2 - 8) = (5x^2)^2 - 8^2 = 25x^4 - 64
\][/tex]
This expression is equivalent to [tex]\(25x^4 - 64\)[/tex].
4. Fourth Expression: [tex]\((x^2 + 13)(x^2 - 13)\)[/tex]
This follows the same difference of squares formula:
[tex]\[
(x^2 + 13)(x^2 - 13) = (x^2)^2 - 13^2 = x^4 - 169
\][/tex]
This simplifies to [tex]\(x^4 - 169\)[/tex], which is not equivalent to [tex]\(25x^4 - 64\)[/tex].
5. Fifth Expression: [tex]\((5x^2 - 8)^2\)[/tex]
Expanding this, we get:
[tex]\[
(5x^2 - 8)^2 = (5x^2)^2 - 2 \times 5x^2 \times 8 + 8^2 = 25x^4 - 80x^2 + 64
\][/tex]
This is not equivalent to [tex]\(25x^4 - 64\)[/tex].
With these evaluations, the expressions equivalent to [tex]\(25x^4 - 64\)[/tex] are:
1. [tex]\(25x^4 + 40x - 40x - 64\)[/tex]
2. [tex]\(25x^4 + 13x - 13x - 64\)[/tex]
3. [tex]\((5x^2 + 8)(5x^2 - 8)\)[/tex]
These are the expressions that match.
1. First Expression: [tex]\(25x^4 + 40x - 40x - 64\)[/tex]
To find if this expression is equivalent to [tex]\(25x^4 - 64\)[/tex], simplify it:
[tex]\[
25x^4 + 40x - 40x - 64 = 25x^4 + (40x - 40x) - 64 = 25x^4 - 64
\][/tex]
This expression is equivalent to [tex]\(25x^4 - 64\)[/tex].
2. Second Expression: [tex]\(25x^4 + 13x - 13x - 64\)[/tex]
Simplify this expression:
[tex]\[
25x^4 + 13x - 13x - 64 = 25x^4 + (13x - 13x) - 64 = 25x^4 - 64
\][/tex]
This expression is also equivalent to [tex]\(25x^4 - 64\)[/tex].
3. Third Expression: [tex]\((5x^2 + 8)(5x^2 - 8)\)[/tex]
This expression is in the form of a difference of squares, which states:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Here, [tex]\(a = 5x^2\)[/tex] and [tex]\(b = 8\)[/tex], therefore:
[tex]\[
(5x^2 + 8)(5x^2 - 8) = (5x^2)^2 - 8^2 = 25x^4 - 64
\][/tex]
This expression is equivalent to [tex]\(25x^4 - 64\)[/tex].
4. Fourth Expression: [tex]\((x^2 + 13)(x^2 - 13)\)[/tex]
This follows the same difference of squares formula:
[tex]\[
(x^2 + 13)(x^2 - 13) = (x^2)^2 - 13^2 = x^4 - 169
\][/tex]
This simplifies to [tex]\(x^4 - 169\)[/tex], which is not equivalent to [tex]\(25x^4 - 64\)[/tex].
5. Fifth Expression: [tex]\((5x^2 - 8)^2\)[/tex]
Expanding this, we get:
[tex]\[
(5x^2 - 8)^2 = (5x^2)^2 - 2 \times 5x^2 \times 8 + 8^2 = 25x^4 - 80x^2 + 64
\][/tex]
This is not equivalent to [tex]\(25x^4 - 64\)[/tex].
With these evaluations, the expressions equivalent to [tex]\(25x^4 - 64\)[/tex] are:
1. [tex]\(25x^4 + 40x - 40x - 64\)[/tex]
2. [tex]\(25x^4 + 13x - 13x - 64\)[/tex]
3. [tex]\((5x^2 + 8)(5x^2 - 8)\)[/tex]
These are the expressions that match.
