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Sagot :
To determine which of the given algebraic expressions represent a difference of squares, we need to recall that a difference of squares takes the form [tex]\( A^2 - B^2 \)[/tex], where both [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are perfect squares.
Let's analyze each expression individually:
### 1. Expression: [tex]\( 9m^4 - 49n^6 \)[/tex]
- Identify components: [tex]\( 9m^4 \)[/tex] and [tex]\( 49n^6 \)[/tex]
- Check if each term is a perfect square:
- [tex]\( 9m^4 \)[/tex] can be rewritten as [tex]\( (3m^2)^2 \)[/tex]
- [tex]\( 49n^6 \)[/tex] can be rewritten as [tex]\( (7n^3)^2 \)[/tex]
Both terms are perfect squares (since [tex]\( 9m^4 = (3m^2)^2 \)[/tex] and [tex]\( 49n^6 = (7n^3)^2 \)[/tex]), so this expression is a difference of squares.
### 2. Expression: [tex]\( 32a^2 - 81n^2 \)[/tex]
- Identify components: [tex]\( 32a^2 \)[/tex] and [tex]\( 81n^2 \)[/tex]
- Check if each term is a perfect square:
- [tex]\( 32a^2 \)[/tex] is not a perfect square because 32 is not a perfect square (since [tex]\(\sqrt{32}\)[/tex] is not an integer)
- [tex]\( 81n^2 \)[/tex] can be rewritten as [tex]\( (9n)^2 \)[/tex], which is a perfect square
Since [tex]\( 32a^2 \)[/tex] is not a perfect square, this expression is not a difference of squares.
### 3. Expression: [tex]\( 16h^2 - 21t^{10} \)[/tex]
- Identify components: [tex]\( 16h^2 \)[/tex] and [tex]\( 21t^{10} \)[/tex]
- Check if each term is a perfect square:
- [tex]\( 16h^2 \)[/tex] can be rewritten as [tex]\( (4h)^2 \)[/tex]
- [tex]\( 21t^{10} \)[/tex] is not a perfect square because 21 is not a perfect square (since [tex]\(\sqrt{21}\)[/tex] is not an integer)
Since [tex]\( 21t^{10} \)[/tex] is not a perfect square, this expression is not a difference of squares.
### 4. Expression: [tex]\( 100x^2 - 10y^4 \)[/tex]
- Identify components: [tex]\( 100x^2 \)[/tex] and [tex]\( 10y^4 \)[/tex]
- Check if each term is a perfect square:
- [tex]\( 100x^2 \)[/tex] can be rewritten as [tex]\( (10x)^2 \)[/tex]
- [tex]\( 10y^4 \)[/tex] is not a perfect square because 10 is not a perfect square (since [tex]\(\sqrt{10}\)[/tex] is not an integer)
Since [tex]\( 10y^4 \)[/tex] is not a perfect square, this expression is not a difference of squares.
### Conclusion:
After analyzing each expression, we find that only the first expression [tex]\( 9m^4 - 49n^6 \)[/tex] is a difference of squares.
Hence, the expression that is a difference of squares is:
[tex]\[ 9m^4 - 49n^6 \][/tex]
Let's analyze each expression individually:
### 1. Expression: [tex]\( 9m^4 - 49n^6 \)[/tex]
- Identify components: [tex]\( 9m^4 \)[/tex] and [tex]\( 49n^6 \)[/tex]
- Check if each term is a perfect square:
- [tex]\( 9m^4 \)[/tex] can be rewritten as [tex]\( (3m^2)^2 \)[/tex]
- [tex]\( 49n^6 \)[/tex] can be rewritten as [tex]\( (7n^3)^2 \)[/tex]
Both terms are perfect squares (since [tex]\( 9m^4 = (3m^2)^2 \)[/tex] and [tex]\( 49n^6 = (7n^3)^2 \)[/tex]), so this expression is a difference of squares.
### 2. Expression: [tex]\( 32a^2 - 81n^2 \)[/tex]
- Identify components: [tex]\( 32a^2 \)[/tex] and [tex]\( 81n^2 \)[/tex]
- Check if each term is a perfect square:
- [tex]\( 32a^2 \)[/tex] is not a perfect square because 32 is not a perfect square (since [tex]\(\sqrt{32}\)[/tex] is not an integer)
- [tex]\( 81n^2 \)[/tex] can be rewritten as [tex]\( (9n)^2 \)[/tex], which is a perfect square
Since [tex]\( 32a^2 \)[/tex] is not a perfect square, this expression is not a difference of squares.
### 3. Expression: [tex]\( 16h^2 - 21t^{10} \)[/tex]
- Identify components: [tex]\( 16h^2 \)[/tex] and [tex]\( 21t^{10} \)[/tex]
- Check if each term is a perfect square:
- [tex]\( 16h^2 \)[/tex] can be rewritten as [tex]\( (4h)^2 \)[/tex]
- [tex]\( 21t^{10} \)[/tex] is not a perfect square because 21 is not a perfect square (since [tex]\(\sqrt{21}\)[/tex] is not an integer)
Since [tex]\( 21t^{10} \)[/tex] is not a perfect square, this expression is not a difference of squares.
### 4. Expression: [tex]\( 100x^2 - 10y^4 \)[/tex]
- Identify components: [tex]\( 100x^2 \)[/tex] and [tex]\( 10y^4 \)[/tex]
- Check if each term is a perfect square:
- [tex]\( 100x^2 \)[/tex] can be rewritten as [tex]\( (10x)^2 \)[/tex]
- [tex]\( 10y^4 \)[/tex] is not a perfect square because 10 is not a perfect square (since [tex]\(\sqrt{10}\)[/tex] is not an integer)
Since [tex]\( 10y^4 \)[/tex] is not a perfect square, this expression is not a difference of squares.
### Conclusion:
After analyzing each expression, we find that only the first expression [tex]\( 9m^4 - 49n^6 \)[/tex] is a difference of squares.
Hence, the expression that is a difference of squares is:
[tex]\[ 9m^4 - 49n^6 \][/tex]
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