Answer :
To find out which of the expressions represents a "difference of squares," we first need to understand what a "difference of squares" is. A difference of squares is a mathematical expression that can be written in the form [tex]\( a^2 - b^2 \)[/tex], where both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are squares.
Let's analyze each of the given expressions to see which one fits this pattern:
1. [tex]\( x^4 + 9y^2 \)[/tex]
- This is a sum of squares, not a difference. Therefore, it does not fit the form [tex]\( a^2 - b^2 \)[/tex].
2. [tex]\( x^5 + 9y^2 \)[/tex]
- This expression is also a sum and not a difference, and [tex]\( x^5 \)[/tex] is not a perfect square, since it's not an even power of [tex]\( x \)[/tex].
3. [tex]\( x^5 - 9y^2 \)[/tex]
- While this is a difference, [tex]\( x^5 \)[/tex] is not a perfect square because the exponent 5 is not even.
4. [tex]\( x^4 - 9y^2 \)[/tex]
- [tex]\( x^4 \)[/tex] is a perfect square because it can be written as [tex]\((x^2)^2\)[/tex].
- [tex]\( 9y^2 \)[/tex] is a perfect square because it can be written as [tex]\((3y)^2\)[/tex].
- Thus, this expression can be written in the form [tex]\( (x^2)^2 - (3y)^2 \)[/tex], which matches the pattern of a difference of squares, [tex]\( a^2 - b^2 \)[/tex].
Therefore, the expression [tex]\( x^4 - 9y^2 \)[/tex] is the only one that represents a "difference of squares."
Let's analyze each of the given expressions to see which one fits this pattern:
1. [tex]\( x^4 + 9y^2 \)[/tex]
- This is a sum of squares, not a difference. Therefore, it does not fit the form [tex]\( a^2 - b^2 \)[/tex].
2. [tex]\( x^5 + 9y^2 \)[/tex]
- This expression is also a sum and not a difference, and [tex]\( x^5 \)[/tex] is not a perfect square, since it's not an even power of [tex]\( x \)[/tex].
3. [tex]\( x^5 - 9y^2 \)[/tex]
- While this is a difference, [tex]\( x^5 \)[/tex] is not a perfect square because the exponent 5 is not even.
4. [tex]\( x^4 - 9y^2 \)[/tex]
- [tex]\( x^4 \)[/tex] is a perfect square because it can be written as [tex]\((x^2)^2\)[/tex].
- [tex]\( 9y^2 \)[/tex] is a perfect square because it can be written as [tex]\((3y)^2\)[/tex].
- Thus, this expression can be written in the form [tex]\( (x^2)^2 - (3y)^2 \)[/tex], which matches the pattern of a difference of squares, [tex]\( a^2 - b^2 \)[/tex].
Therefore, the expression [tex]\( x^4 - 9y^2 \)[/tex] is the only one that represents a "difference of squares."
