Answer :
We want to know which of the following expressions simplify to
[tex]$$
25x^4 - 64.
$$[/tex]
Let's analyze each option step by step.
1. Option 1:
[tex]$$
25x^4 + 40x - 40x - 64.
$$[/tex]
The terms [tex]$+40x$[/tex] and [tex]$-40x$[/tex] cancel each other, so we have
[tex]$$
25x^4 - 64.
$$[/tex]
This is exactly the target expression.
2. Option 2:
[tex]$$
25x^4 + 13x - 13x - 64.
$$[/tex]
Again, the terms [tex]$+13x$[/tex] and [tex]$-13x$[/tex] cancel, leaving us with
[tex]$$
25x^4 - 64.
$$[/tex]
This matches the target expression as well.
3. Option 3:
[tex]$$
(5x^2+8)(5x^2-8).
$$[/tex]
This product is in the form of a difference of two squares:
[tex]$$
(5x^2)^2 - 8^2 = 25x^4 - 64.
$$[/tex]
Hence, this expression is equivalent to the target.
4. Option 4:
[tex]$$
(x^2+13)(x^2-13).
$$[/tex]
This is also a difference of two squares:
[tex]$$
(x^2)^2 - 13^2 = x^4 - 169.
$$[/tex]
Since [tex]$x^4 - 169$[/tex] is not the same as [tex]$25x^4 - 64$[/tex], this option is not equivalent.
5. Option 5:
[tex]$$
(5x^2-8)^2.
$$[/tex]
Expanding this square gives:
[tex]$$
(5x^2)^2 - 2\cdot 5x^2\cdot 8 + 8^2 = 25x^4 - 80x^2 + 64.
$$[/tex]
This result is clearly different from [tex]$25x^4 - 64$[/tex], so it is not equivalent.
Thus, the expressions equivalent to [tex]$25x^4-64$[/tex] are those given in Options 1, 2, and 3.
[tex]$$
25x^4 - 64.
$$[/tex]
Let's analyze each option step by step.
1. Option 1:
[tex]$$
25x^4 + 40x - 40x - 64.
$$[/tex]
The terms [tex]$+40x$[/tex] and [tex]$-40x$[/tex] cancel each other, so we have
[tex]$$
25x^4 - 64.
$$[/tex]
This is exactly the target expression.
2. Option 2:
[tex]$$
25x^4 + 13x - 13x - 64.
$$[/tex]
Again, the terms [tex]$+13x$[/tex] and [tex]$-13x$[/tex] cancel, leaving us with
[tex]$$
25x^4 - 64.
$$[/tex]
This matches the target expression as well.
3. Option 3:
[tex]$$
(5x^2+8)(5x^2-8).
$$[/tex]
This product is in the form of a difference of two squares:
[tex]$$
(5x^2)^2 - 8^2 = 25x^4 - 64.
$$[/tex]
Hence, this expression is equivalent to the target.
4. Option 4:
[tex]$$
(x^2+13)(x^2-13).
$$[/tex]
This is also a difference of two squares:
[tex]$$
(x^2)^2 - 13^2 = x^4 - 169.
$$[/tex]
Since [tex]$x^4 - 169$[/tex] is not the same as [tex]$25x^4 - 64$[/tex], this option is not equivalent.
5. Option 5:
[tex]$$
(5x^2-8)^2.
$$[/tex]
Expanding this square gives:
[tex]$$
(5x^2)^2 - 2\cdot 5x^2\cdot 8 + 8^2 = 25x^4 - 80x^2 + 64.
$$[/tex]
This result is clearly different from [tex]$25x^4 - 64$[/tex], so it is not equivalent.
Thus, the expressions equivalent to [tex]$25x^4-64$[/tex] are those given in Options 1, 2, and 3.
