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Answer :
Sure! Let's analyze each given expression to see which ones are equivalent to [tex]\( 25x^4 - 64 \)[/tex].
1. Expression: [tex]\( 25x^4 + 40x - 40x - 64 \)[/tex]
Let's simplify this expression step-by-step:
[tex]\[
25x^4 + 40x - 40x - 64
\][/tex]
Combine like terms (the [tex]\( +40x \)[/tex] and [tex]\( -40x \)[/tex] cancel each other out):
[tex]\[
25x^4 - 64
\][/tex]
This is exactly [tex]\( 25x^4 - 64 \)[/tex], so this is equivalent.
2. Expression: [tex]\( 25x^4 + 13x - 13x - 64 \)[/tex]
Simplify this expression step-by-step:
[tex]\[
25x^4 + 13x - 13x - 64
\][/tex]
Combine like terms (the [tex]\( +13x \)[/tex] and [tex]\( -13x \)[/tex] cancel each other out):
[tex]\[
25x^4 - 64
\][/tex]
This is exactly [tex]\( 25x^4 - 64 \)[/tex], so this is equivalent.
3. Expression: [tex]\( (5x^2 + 8)(5x^2 - 8) \)[/tex]
Expand this using the difference of squares formula [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex]:
[tex]\[
(5x^2 + 8)(5x^2 - 8) = (5x^2)^2 - 8^2 = 25x^4 - 64
\][/tex]
This is exactly [tex]\( 25x^4 - 64 \)[/tex], so this is equivalent.
4. Expression: [tex]\( (x^2 + 13)(x^2 - 13) \)[/tex]
Expand this using the difference of squares formula [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex]:
[tex]\[
(x^2 + 13)(x^2 - 13) = (x^2)^2 - 13^2 = x^4 - 169
\][/tex]
This is [tex]\( x^4 - 169 \)[/tex], which is not the same as [tex]\( 25x^4 - 64 \)[/tex], so this is not equivalent.
5. Expression: [tex]\( (5x^2 - 8)^2 \)[/tex]
Expand this using the square of a binomial formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(5x^2 - 8)^2 = (5x^2)^2 - 2 \cdot 5x^2 \cdot 8 + 8^2 = 25x^4 - 80x^2 + 64
\][/tex]
This is [tex]\( 25x^4 - 80x^2 + 64 \)[/tex], which is not the same as [tex]\( 25x^4 - 64 \)[/tex], so this is not equivalent.
Given these detailed steps, the three expressions that are equivalent to [tex]\( 25x^4 - 64 \)[/tex] are:
- [tex]\( 25x^4 + 40x - 40x - 64 \)[/tex]
- [tex]\( 25x^4 + 13x - 13x - 64 \)[/tex]
- [tex]\( (5x^2 + 8)(5x^2 - 8) \)[/tex]
So the correct options are the first, second, and third expressions.
1. Expression: [tex]\( 25x^4 + 40x - 40x - 64 \)[/tex]
Let's simplify this expression step-by-step:
[tex]\[
25x^4 + 40x - 40x - 64
\][/tex]
Combine like terms (the [tex]\( +40x \)[/tex] and [tex]\( -40x \)[/tex] cancel each other out):
[tex]\[
25x^4 - 64
\][/tex]
This is exactly [tex]\( 25x^4 - 64 \)[/tex], so this is equivalent.
2. Expression: [tex]\( 25x^4 + 13x - 13x - 64 \)[/tex]
Simplify this expression step-by-step:
[tex]\[
25x^4 + 13x - 13x - 64
\][/tex]
Combine like terms (the [tex]\( +13x \)[/tex] and [tex]\( -13x \)[/tex] cancel each other out):
[tex]\[
25x^4 - 64
\][/tex]
This is exactly [tex]\( 25x^4 - 64 \)[/tex], so this is equivalent.
3. Expression: [tex]\( (5x^2 + 8)(5x^2 - 8) \)[/tex]
Expand this using the difference of squares formula [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex]:
[tex]\[
(5x^2 + 8)(5x^2 - 8) = (5x^2)^2 - 8^2 = 25x^4 - 64
\][/tex]
This is exactly [tex]\( 25x^4 - 64 \)[/tex], so this is equivalent.
4. Expression: [tex]\( (x^2 + 13)(x^2 - 13) \)[/tex]
Expand this using the difference of squares formula [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex]:
[tex]\[
(x^2 + 13)(x^2 - 13) = (x^2)^2 - 13^2 = x^4 - 169
\][/tex]
This is [tex]\( x^4 - 169 \)[/tex], which is not the same as [tex]\( 25x^4 - 64 \)[/tex], so this is not equivalent.
5. Expression: [tex]\( (5x^2 - 8)^2 \)[/tex]
Expand this using the square of a binomial formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(5x^2 - 8)^2 = (5x^2)^2 - 2 \cdot 5x^2 \cdot 8 + 8^2 = 25x^4 - 80x^2 + 64
\][/tex]
This is [tex]\( 25x^4 - 80x^2 + 64 \)[/tex], which is not the same as [tex]\( 25x^4 - 64 \)[/tex], so this is not equivalent.
Given these detailed steps, the three expressions that are equivalent to [tex]\( 25x^4 - 64 \)[/tex] are:
- [tex]\( 25x^4 + 40x - 40x - 64 \)[/tex]
- [tex]\( 25x^4 + 13x - 13x - 64 \)[/tex]
- [tex]\( (5x^2 + 8)(5x^2 - 8) \)[/tex]
So the correct options are the first, second, and third expressions.
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