Álgebra Ejemplos

{(2 x^{2} - 3 y)}^{3}

${(2 x^{2} - 3 y)}^{3}$

Paso 1

El triángulo de Pascal se puede visualizar de la siguiente manera:

1

$1$

1 - 1

$1 - 1$

1 - 2 - 1

$1 - 2 - 1$

1 - 3 - 3 - 1

$1 - 3 - 3 - 1$

El triángulo puede usarse para calcular los coeficientes de la expansión de

{(a + b)}^{n}

${(a + b)}^{n}$ al tomar el exponente

n

$n$ y sumar

1

$1$ . Los coeficientes se corresponderán con la línea

n + 1

$n + 1$ del triángulo. Para

{(2 x^{2} - 3 y)}^{3}

${(2 x^{2} - 3 y)}^{3}$ ,

n = 3

$n = 3$ de modo que los coeficientes de la expansión se corresponderán con la línea

4

$4$ .

Paso 2

La expansión sigue la regla

{(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}

${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . Los valores de los coeficientes, desde el triángulo, son

1 - 3 - 3 - 1

$1 - 3 - 3 - 1$ .

1 a^{3} b^{0} + 3 a^{2} b + 3 a b^{2} + 1 a^{0} b^{3}

$1 a^{3} b^{0} + 3 a^{2} b + 3 a b^{2} + 1 a^{0} b^{3}$

Paso 3

Sustituye los valores reales de

a

$a$ ,

2 x^{2}

$2 x^{2}$ y

b

$b$ en la expresión

- 3 y

$- 3 y$ .

1 {(2 x^{2})}^{3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$1 {(2 x^{2})}^{3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4

Simplifica cada término.

Toca para ver más pasos...

Paso 4.1

Multiplica

{(2 x^{2})}^{3}

${(2 x^{2})}^{3}$ por

1

$1$ .

{(2 x^{2})}^{3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

${(2 x^{2})}^{3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.2

Aplica la regla del producto a

2 x^{2}

$2 x^{2}$ .

2^{3} {(x^{2})}^{3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$2^{3} {(x^{2})}^{3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.3

Eleva

2

$2$ a la potencia de

3

$3$ .

8 {(x^{2})}^{3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 {(x^{2})}^{3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.4

Multiplica los exponentes en

{(x^{2})}^{3}

${(x^{2})}^{3}$ .

Toca para ver más pasos...

Paso 4.4.1

Aplica la regla de la potencia y multiplica los exponentes,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

8 x^{2 \cdot 3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{2 \cdot 3} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.4.2

Multiplica

2

$2$ por

3

$3$ .

8 x^{6} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

8 x^{6} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} {(- 3 y)}^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.5

Aplica la regla del producto a

- 3 y

$- 3 y$ .

8 x^{6} ({(- 3)}^{0} y^{0}) + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} ({(- 3)}^{0} y^{0}) + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.6

Reescribe con la propiedad conmutativa de la multiplicación.

8 \cdot {(- 3)}^{0} x^{6} y^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 \cdot {(- 3)}^{0} x^{6} y^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.7

Cualquier valor elevado a

0

$0$ es

1

$1$ .

8 \cdot 1 x^{6} y^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 \cdot 1 x^{6} y^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.8

Multiplica

8

$8$ por

1

$1$ .

8 x^{6} y^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} y^{0} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.9

Cualquier valor elevado a

0

$0$ es

1

$1$ .

8 x^{6} \cdot 1 + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} \cdot 1 + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.10

Multiplica

8

$8$ por

1

$1$ .

8 x^{6} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 3 {(2 x^{2})}^{2} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.11

Aplica la regla del producto a

2 x^{2}

$2 x^{2}$ .

8 x^{6} + 3 (2^{2} {(x^{2})}^{2}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 3 (2^{2} {(x^{2})}^{2}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.12

Eleva

2

$2$ a la potencia de

2

$2$ .

8 x^{6} + 3 (4 {(x^{2})}^{2}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 3 (4 {(x^{2})}^{2}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.13

Multiplica los exponentes en

{(x^{2})}^{2}

${(x^{2})}^{2}$ .

Toca para ver más pasos...

Paso 4.13.1

Aplica la regla de la potencia y multiplica los exponentes,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

8 x^{6} + 3 (4 x^{2 \cdot 2}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 3 (4 x^{2 \cdot 2}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.13.2

Multiplica

2

$2$ por

2

$2$ .

8 x^{6} + 3 (4 x^{4}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 3 (4 x^{4}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

8 x^{6} + 3 (4 x^{4}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 3 (4 x^{4}) {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.14

Multiplica

4

$4$ por

3

$3$ .

8 x^{6} + 12 x^{4} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 12 x^{4} {(- 3 y)}^{1} + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.15

Simplifica.

8 x^{6} + 12 x^{4} (- 3 y) + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 12 x^{4} (- 3 y) + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.16

Reescribe con la propiedad conmutativa de la multiplicación.

8 x^{6} + 12 \cdot - 3 x^{4} y + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} + 12 \cdot - 3 x^{4} y + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.17

Multiplica

12

$12$ por

- 3

$- 3$ .

8 x^{6} - 36 x^{4} y + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 3 {(2 x^{2})}^{1} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.18

Simplifica.

8 x^{6} - 36 x^{4} y + 3 (2 x^{2}) {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 3 (2 x^{2}) {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.19

Multiplica

2

$2$ por

3

$3$ .

8 x^{6} - 36 x^{4} y + 6 x^{2} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 6 x^{2} {(- 3 y)}^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.20

Aplica la regla del producto a

- 3 y

$- 3 y$ .

8 x^{6} - 36 x^{4} y + 6 x^{2} ({(- 3)}^{2} y^{2}) + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 6 x^{2} ({(- 3)}^{2} y^{2}) + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.21

Reescribe con la propiedad conmutativa de la multiplicación.

8 x^{6} - 36 x^{4} y + 6 \cdot {(- 3)}^{2} x^{2} y^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 6 \cdot {(- 3)}^{2} x^{2} y^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.22

Eleva

- 3

$- 3$ a la potencia de

2

$2$ .

8 x^{6} - 36 x^{4} y + 6 \cdot 9 x^{2} y^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 6 \cdot 9 x^{2} y^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.23

Multiplica

6

$6$ por

9

$9$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + 1 {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.24

Multiplica

{(2 x^{2})}^{0}

${(2 x^{2})}^{0}$ por

1

$1$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + {(2 x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + {(2 x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.25

Aplica la regla del producto a

2 x^{2}

$2 x^{2}$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + 2^{0} {(x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + 2^{0} {(x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.26

Cualquier valor elevado a

0

$0$ es

1

$1$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + 1 {(x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + 1 {(x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.27

Multiplica

{(x^{2})}^{0}

${(x^{2})}^{0}$ por

1

$1$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + {(x^{2})}^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + {(x^{2})}^{0} {(- 3 y)}^{3}$

Paso 4.28

Multiplica los exponentes en

{(x^{2})}^{0}

${(x^{2})}^{0}$ .

Toca para ver más pasos...

Paso 4.28.1

Aplica la regla de la potencia y multiplica los exponentes,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + x^{2 \cdot 0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + x^{2 \cdot 0} {(- 3 y)}^{3}$

Paso 4.28.2

Multiplica

2

$2$ por

0

$0$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + x^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + x^{0} {(- 3 y)}^{3}$

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + x^{0} {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + x^{0} {(- 3 y)}^{3}$

Paso 4.29

Cualquier valor elevado a

0

$0$ es

1

$1$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + 1 {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + 1 {(- 3 y)}^{3}$

Paso 4.30

Multiplica

{(- 3 y)}^{3}

${(- 3 y)}^{3}$ por

1

$1$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + {(- 3 y)}^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + {(- 3 y)}^{3}$

Paso 4.31

Aplica la regla del producto a

- 3 y

$- 3 y$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + {(- 3)}^{3} y^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} + {(- 3)}^{3} y^{3}$

Paso 4.32

Eleva

- 3

$- 3$ a la potencia de

3

$3$ .

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} - 27 y^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} - 27 y^{3}$

8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} - 27 y^{3}

$8 x^{6} - 36 x^{4} y + 54 x^{2} y^{2} - 27 y^{3}$