Álgebra Ejemplos

${(x^{- 1} + 2 y^{- 1})}^{4}$

Paso 1

Reescribe la expresión mediante la regla del exponente negativo $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{x} + 2 y^{- 1})}^{4}$

Paso 2

Reescribe la expresión mediante la regla del exponente negativo $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{x} + 2 (\frac{1}{y}))}^{4}$

Paso 3

Combina $2$ y $\frac{1}{y}$ .

${(\frac{1}{x} + \frac{2}{y})}^{4}$

Paso 4

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

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

El triángulo puede usarse para calcular los coeficientes de la expansión de ${(a + b)}^{n}$ al tomar el exponente $n$ y sumar $1$ . Los coeficientes se corresponderán con la línea $n + 1$ del triángulo. Para ${(\frac{1}{x} + \frac{2}{y})}^{4}$ , $n = 4$ de modo que los coeficientes de la expansión se corresponderán con la línea $5$ .

Paso 5

La expansión sigue la regla ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . Los valores de los coeficientes, desde el triángulo, son $1 - 4 - 6 - 4 - 1$ .

${(1 a^{4} b^{0} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 a^{0} b^{4})}^{4}$

Paso 6

Sustituye los valores reales de $a$ , $\frac{1}{x}$ y $b$ en la expresión $\frac{2}{y}$ .

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

Paso 7

Simplifica cada término.

Toca para ver más pasos...

Paso 7.1

Multiplica ${(\frac{1}{x})}^{4}$ por $1$ .

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

Paso 7.2

Aplica la regla del producto a $\frac{1}{x}$ .

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

Paso 7.3

Uno elevado a cualquier potencia es uno.

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

Paso 7.4

Aplica la regla del producto a $\frac{2}{y}$ .

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

Paso 7.5

Combinar.

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

Paso 7.6

Multiplica $2^{0}$ por $1$ .

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

Paso 7.7

Cualquier valor elevado a $0$ es $1$ .

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

Paso 7.8

Cualquier valor elevado a $0$ es $1$ .

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

Paso 7.9

Multiplica $x^{4}$ por $1$ .

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

Paso 7.10

Aplica la regla del producto a $\frac{1}{x}$ .

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

Paso 7.11

Uno elevado a cualquier potencia es uno.

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

Paso 7.12

Combina $4$ y $\frac{1}{x^{3}}$ .

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

Paso 7.13

Simplifica.

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

Paso 7.14

Combinar.

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

Paso 7.15

Multiplica $4$ por $2$ .

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

Paso 7.16

Aplica la regla del producto a $\frac{1}{x}$ .

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

Paso 7.17

Uno elevado a cualquier potencia es uno.

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

Paso 7.18

Combina $6$ y $\frac{1}{x^{2}}$ .

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

Paso 7.19

Aplica la regla del producto a $\frac{2}{y}$ .

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

Paso 7.20

Combinar.

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

Paso 7.21

Eleva $2$ a la potencia de $2$ .

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

Paso 7.22

Multiplica $6$ por $4$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + 4 (\frac{1}{x}) {(\frac{2}{y})}^{3} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.23

Simplifica.

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + 4 (\frac{1}{x}) \cdot {(\frac{2}{y})}^{3} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.24

Combina $4$ y $\frac{1}{x}$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{4}{x} \cdot {(\frac{2}{y})}^{3} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.25

Aplica la regla del producto a $\frac{2}{y}$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{4}{x} \cdot \frac{2^{3}}{y^{3}} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.26

Combinar.

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{4 \cdot 2^{3}}{x y^{3}} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.27

Simplifica el numerador.

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Paso 7.27.1

Reescribe $4$ como $2^{2}$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{2^{2} \cdot 2^{3}}{x y^{3}} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.27.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{2^{2 + 3}}{x y^{3}} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.27.3

Suma $2$ y $3$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{2^{5}}{x y^{3}} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.28

Eleva $2$ a la potencia de $5$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + 1 {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.29

Multiplica ${(\frac{1}{x})}^{0}$ por $1$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + {(\frac{1}{x})}^{0} {(\frac{2}{y})}^{4})}^{4}$

Paso 7.30

Aplica la regla del producto a $\frac{1}{x}$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + \frac{1^{0}}{x^{0}} \cdot {(\frac{2}{y})}^{4})}^{4}$

Paso 7.31

Cualquier valor elevado a $0$ es $1$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + \frac{1}{x^{0}} \cdot {(\frac{2}{y})}^{4})}^{4}$

Paso 7.32

Cualquier valor elevado a $0$ es $1$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + \frac{1}{1} \cdot {(\frac{2}{y})}^{4})}^{4}$

Paso 7.33

Divide $1$ por $1$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + 1 {(\frac{2}{y})}^{4})}^{4}$

Paso 7.34

Multiplica ${(\frac{2}{y})}^{4}$ por $1$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + {(\frac{2}{y})}^{4})}^{4}$

Paso 7.35

Aplica la regla del producto a $\frac{2}{y}$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + \frac{2^{4}}{y^{4}})}^{4}$

Paso 7.36

Eleva $2$ a la potencia de $4$ .

${(\frac{1}{x^{4}} + \frac{8}{x^{3} y} + \frac{24}{x^{2} y^{2}} + \frac{32}{x y^{3}} + \frac{16}{y^{4}})}^{4}$