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Expand the following expressions:

a) [tex]\(\left(3 x^2\right)^3\)[/tex]

b) [tex]\((4x + 3)^2\)[/tex]

c) [tex]\((3x + 2y)(3x - 2y)\)[/tex]

d) [tex]\(3 \cdot \left(5x^2 - 1\right)\)[/tex]

e) [tex]\((2 - 5y)^2\)[/tex]

Sagot :

GinnyAnswer
Claro, vamos a expandir paso a paso cada una de las expresiones dadas.

### a) [tex]\(\left(3 x^2\right)^3\)[/tex]

Primero, analizamos la expresión [tex]\(\left(3 x^2\right)^3\)[/tex].

- Elevamos cada término dentro del paréntesis al cubo:
[tex]\[ (3 x^2)^3 = 3^3 \cdot (x^2)^3 \][/tex]
- Resolviendo las potencias, obtenemos:
[tex]\[ 3^3 = 27 \quad \text{y} \quad (x^2)^3 = x^{2 \cdot 3} = x^6 \][/tex]
- El resultado es:
[tex]\[ 27 x^6 \][/tex]

### b) [tex]\((4 x + 3)^2\)[/tex]

Para expandir [tex]\((4 x + 3)^2\)[/tex], usamos el cuadrado del binomio [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]:

- Aquí, [tex]\(a = 4x\)[/tex] y [tex]\(b = 3\)[/tex]:
[tex]\[ (4 x + 3)^2 = (4 x)^2 + 2(4 x)(3) + 3^2 \][/tex]
- Resolviendo cada término, obtenemos:
[tex]\[ (4 x)^2 = 16 x^2, \quad 2(4 x)(3) = 24 x, \quad \text{y} \quad 3^2 = 9 \][/tex]
- El resultado es:
[tex]\[ 16 x^2 + 24 x + 9 \][/tex]

### c) [tex]\((3 x + 2 y)(3 x - 2 y)\)[/tex]

Para expandir [tex]\((3 x + 2 y)(3 x - 2 y)\)[/tex], reconocemos el producto de dos binomios conjugados: [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]:

- Aquí, [tex]\(a = 3x\)[/tex] y [tex]\(b = 2y\)[/tex]:
[tex]\[ (3 x + 2 y)(3 x - 2 y) = (3 x)^2 - (2 y)^2 \][/tex]
- Resolviendo cada término, obtenemos:
[tex]\[ (3 x)^2 = 9 x^2 \quad \text{y} \quad (2 y)^2 = 4 y^2 \][/tex]
- El resultado es:
[tex]\[ 9 x^2 - 4 y^2 \][/tex]

### d) [tex]\(3 \cdot (5 x^2 - 1)\)[/tex]

Para expandir [tex]\(3 \cdot (5 x^2 - 1)\)[/tex], distribuimos el [tex]\(3\)[/tex] a cada término dentro del paréntesis:

- Multiplicamos [tex]\(3\)[/tex] por cada término:
[tex]\[ 3 \cdot (5 x^2 - 1) = 3 \cdot 5 x^2 + 3 \cdot (-1) \][/tex]
- Resolviendo las multiplicaciones, obtenemos:
[tex]\[ 15 x^2 - 3 \][/tex]

### e) [tex]\((2 - 5 y)^2\)[/tex]

Para expandir [tex]\((2 - 5 y)^2\)[/tex], usamos el cuadrado del binomio [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:

- Aquí, [tex]\(a = 2\)[/tex] y [tex]\(b = 5y\)[/tex]:
[tex]\[ (2 - 5 y)^2 = 2^2 - 2(2)(5 y) + (5 y)^2 \][/tex]
- Resolviendo cada término, obtenemos:
[tex]\[ 2^2 = 4, \quad -2(2)(5 y) = -20 y, \quad \text{y} \quad (5 y)^2 = 25 y^2 \][/tex]
- El resultado es:
[tex]\[ 4 - 20 y + 25 y^2 \][/tex]

Entonces, los resultados de las expansiones son:

a) [tex]\(27 x^6\)[/tex]
b) [tex]\(16 x^2 + 24 x + 9\)[/tex]
c) [tex]\(9 x^2 - 4 y^2\)[/tex]
d) [tex]\(15 x^2 - 3\)[/tex]
e) [tex]\(4 - 20 y + 25 y^2\)[/tex]
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