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Sagot :
Para realizar los productos indicados, debemos expandir cada uno de los binomios cuadrados. A continuación se muestran las soluciones paso a paso para cada expresión:
1. Expandir [tex]\((9d - 5f)^2\)[/tex]:
[tex]\[ (9d - 5f)^2 = 81d^2 - 2 \cdot 9d \cdot 5f + 25f^2 = 81d^2 - 90df + 25f^2 \][/tex]
2. Expandir [tex]\((21h + 13m)^2\)[/tex]:
[tex]\[ (21h + 13m)^2 = 441h^2 + 2 \cdot 21h \cdot 13m + 169m^2 = 441h^2 + 546hm + 169m^2 \][/tex]
3. Expandir [tex]\((13a + 10b)^2\)[/tex]:
[tex]\[ (13a + 10b)^2 = 169a^2 + 2 \cdot 13a \cdot 10b + 100b^2 = 169a^2 + 260ab + 100b^2 \][/tex]
4. Expandir [tex]\(\left(2m^5 + 15n\right)^2\)[/tex]:
[tex]\[ (2m^5 + 15n)^2 = (2m^5)^2 + 2 \cdot 2m^5 \cdot 15n + (15n)^2 = 4m^{10} + 60m^5n + 225n^2 \][/tex]
5. Expandir [tex]\(\left(4x^5 + 7y^4\right)^2\)[/tex]:
[tex]\[ (4x^5 + 7y^4)^2 = (4x^5)^2 + 2 \cdot 4x^5 \cdot 7y^4 + (7y^4)^2 = 16x^{10} + 56x^5 y^4 + 49y^8 \][/tex]
6. Expandir [tex]\(\left(12k^8 - 23p^{11}\right)^2\)[/tex]:
[tex]\[ (12k^8 - 23p^{11})^2 = (12k^8)^2 - 2 \cdot 12k^8 \cdot 23p^{11} + (23p^{11})^2 = 144k^{16} - 552k^8 p^{11} + 529p^{22} \][/tex]
7. Expandir [tex]\(\left(x^3 - 8y^9\right)^2\)[/tex]:
[tex]\[ (x^3 - 8y^9)^2 = (x^3)^2 - 2 \cdot x^3 \cdot 8y^9 + (8y^9)^2 = x^6 - 16x^3 y^9 + 64 y^{18} \][/tex]
8. Expandir [tex]\((7a + 11b)^2\)[/tex]:
[tex]\[ (7a + 11b)^2 = 49a^2 + 2 \cdot 7a \cdot 11b + 121b^2 = 49a^2 + 154ab + 121b^2 \][/tex]
9. Expandir [tex]\((17p - 12q)^2\)[/tex]:
[tex]\[ (17p - 12q)^2 = 289p^2 - 2 \cdot 17p \cdot 12q + 144q^2 = 289p^2 - 408pq + 144q^2 \][/tex]
10. Expandir [tex]\((w - z)^2\)[/tex]:
[tex]\[ (w - z)^2 = w^2 - 2wz + z^2 \][/tex]
Después de realizar estas expansiones, obtenemos lo siguiente:
1. [tex]\(81d^2 - 90df + 25f^2\)[/tex]
2. [tex]\(441h^2 + 546hm + 169m^2\)[/tex]
3. [tex]\(169a^2 + 260ab + 100b^2\)[/tex]
4. [tex]\(4m^{10} + 60m^5n + 225n^2\)[/tex]
5. [tex]\(16x^{10} + 56x^5 y^4 + 49y^8\)[/tex]
6. [tex]\(144k^{16} - 552k^8 p^{11} + 529p^{22}\)[/tex]
7. [tex]\(x^6 - 16x^3 y^9 + 64 y^{18}\)[/tex]
8. [tex]\(49a^2 + 154ab + 121b^2\)[/tex]
9. [tex]\(289p^2 - 408pq + 144q^2\)[/tex]
10. [tex]\(w^2 - 2wz + z^2\)[/tex]
Estas son las expresiones finales una vez que los binomios han sido completamente expandidos.
1. Expandir [tex]\((9d - 5f)^2\)[/tex]:
[tex]\[ (9d - 5f)^2 = 81d^2 - 2 \cdot 9d \cdot 5f + 25f^2 = 81d^2 - 90df + 25f^2 \][/tex]
2. Expandir [tex]\((21h + 13m)^2\)[/tex]:
[tex]\[ (21h + 13m)^2 = 441h^2 + 2 \cdot 21h \cdot 13m + 169m^2 = 441h^2 + 546hm + 169m^2 \][/tex]
3. Expandir [tex]\((13a + 10b)^2\)[/tex]:
[tex]\[ (13a + 10b)^2 = 169a^2 + 2 \cdot 13a \cdot 10b + 100b^2 = 169a^2 + 260ab + 100b^2 \][/tex]
4. Expandir [tex]\(\left(2m^5 + 15n\right)^2\)[/tex]:
[tex]\[ (2m^5 + 15n)^2 = (2m^5)^2 + 2 \cdot 2m^5 \cdot 15n + (15n)^2 = 4m^{10} + 60m^5n + 225n^2 \][/tex]
5. Expandir [tex]\(\left(4x^5 + 7y^4\right)^2\)[/tex]:
[tex]\[ (4x^5 + 7y^4)^2 = (4x^5)^2 + 2 \cdot 4x^5 \cdot 7y^4 + (7y^4)^2 = 16x^{10} + 56x^5 y^4 + 49y^8 \][/tex]
6. Expandir [tex]\(\left(12k^8 - 23p^{11}\right)^2\)[/tex]:
[tex]\[ (12k^8 - 23p^{11})^2 = (12k^8)^2 - 2 \cdot 12k^8 \cdot 23p^{11} + (23p^{11})^2 = 144k^{16} - 552k^8 p^{11} + 529p^{22} \][/tex]
7. Expandir [tex]\(\left(x^3 - 8y^9\right)^2\)[/tex]:
[tex]\[ (x^3 - 8y^9)^2 = (x^3)^2 - 2 \cdot x^3 \cdot 8y^9 + (8y^9)^2 = x^6 - 16x^3 y^9 + 64 y^{18} \][/tex]
8. Expandir [tex]\((7a + 11b)^2\)[/tex]:
[tex]\[ (7a + 11b)^2 = 49a^2 + 2 \cdot 7a \cdot 11b + 121b^2 = 49a^2 + 154ab + 121b^2 \][/tex]
9. Expandir [tex]\((17p - 12q)^2\)[/tex]:
[tex]\[ (17p - 12q)^2 = 289p^2 - 2 \cdot 17p \cdot 12q + 144q^2 = 289p^2 - 408pq + 144q^2 \][/tex]
10. Expandir [tex]\((w - z)^2\)[/tex]:
[tex]\[ (w - z)^2 = w^2 - 2wz + z^2 \][/tex]
Después de realizar estas expansiones, obtenemos lo siguiente:
1. [tex]\(81d^2 - 90df + 25f^2\)[/tex]
2. [tex]\(441h^2 + 546hm + 169m^2\)[/tex]
3. [tex]\(169a^2 + 260ab + 100b^2\)[/tex]
4. [tex]\(4m^{10} + 60m^5n + 225n^2\)[/tex]
5. [tex]\(16x^{10} + 56x^5 y^4 + 49y^8\)[/tex]
6. [tex]\(144k^{16} - 552k^8 p^{11} + 529p^{22}\)[/tex]
7. [tex]\(x^6 - 16x^3 y^9 + 64 y^{18}\)[/tex]
8. [tex]\(49a^2 + 154ab + 121b^2\)[/tex]
9. [tex]\(289p^2 - 408pq + 144q^2\)[/tex]
10. [tex]\(w^2 - 2wz + z^2\)[/tex]
Estas son las expresiones finales una vez que los binomios han sido completamente expandidos.
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