Algebra Examples

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

Step 1

Combine $\frac{1}{2}$ and $x$ .

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

Step 2

Use the Binomial Theorem.

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

Step 3

Simplify each term.

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Step 3.1

Apply the product rule to $\frac{x}{2}$ .

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

Step 3.2

Raise $2$ to the power of $4$ .

$\frac{x^{4}}{16} + 4 {(\frac{x}{2})}^{3} (- y) + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.3

Rewrite using the commutative property of multiplication.

$\frac{x^{4}}{16} + 4 \cdot - 1 {(\frac{x}{2})}^{3} y + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.4

Multiply $4$ by $- 1$ .

$\frac{x^{4}}{16} - 4 {(\frac{x}{2})}^{3} y + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.5

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{4}}{16} - 4 \frac{x^{3}}{2^{3}} y + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.6

Raise $2$ to the power of $3$ .

$\frac{x^{4}}{16} - 4 \frac{x^{3}}{8} y + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.7

Cancel the common factor of $4$ .

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Step 3.7.1

Factor $4$ out of $- 4$ .

$\frac{x^{4}}{16} + 4 (- 1) \frac{x^{3}}{8} y + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.7.2

Factor $4$ out of $8$ .

$\frac{x^{4}}{16} + 4 \cdot - 1 \frac{x^{3}}{4 \cdot 2} y + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.7.3

Cancel the common factor.

$\frac{x^{4}}{16} + 4 \cdot - 1 \frac{x^{3}}{4 \cdot 2} y + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.7.4

Rewrite the expression.

$\frac{x^{4}}{16} - \frac{x^{3}}{2} y + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.8

Combine $y$ and $\frac{x^{3}}{2}$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + 6 {(\frac{x}{2})}^{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.9

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + 6 \frac{x^{2}}{2^{2}} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.10

Raise $2$ to the power of $2$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + 6 \frac{x^{2}}{4} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.11

Cancel the common factor of $2$ .

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Step 3.11.1

Factor $2$ out of $6$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + 2 (3) \frac{x^{2}}{4} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.11.2

Factor $2$ out of $4$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + 2 \cdot 3 \frac{x^{2}}{2 \cdot 2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.11.3

Cancel the common factor.

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + 2 \cdot 3 \frac{x^{2}}{2 \cdot 2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.11.4

Rewrite the expression.

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + 3 \frac{x^{2}}{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.12

Combine $3$ and $\frac{x^{2}}{2}$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2}}{2} {(- y)}^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.13

Apply the product rule to $- y$ .

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

Step 3.14

Rewrite using the commutative property of multiplication.

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

Step 3.15

Raise $- 1$ to the power of $2$ .

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

Step 3.16

Multiply $\frac{3 x^{2}}{2}$ by $1$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2}}{2} y^{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.17

Combine $\frac{3 x^{2}}{2}$ and $y^{2}$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} + 4 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.18

Cancel the common factor of $2$ .

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Step 3.18.1

Factor $2$ out of $4$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} + 2 (2) \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.18.2

Cancel the common factor.

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} + 2 \cdot 2 \frac{x}{2} {(- y)}^{3} + {(- y)}^{4}$

Step 3.18.3

Rewrite the expression.

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} + 2 x {(- y)}^{3} + {(- y)}^{4}$

Step 3.19

Apply the product rule to $- y$ .

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

Step 3.20

Rewrite using the commutative property of multiplication.

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} + 2 \cdot {(- 1)}^{3} x y^{3} + {(- y)}^{4}$

Step 3.21

Raise $- 1$ to the power of $3$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} + 2 \cdot - 1 x y^{3} + {(- y)}^{4}$

Step 3.22

Multiply $2$ by $- 1$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} - 2 x y^{3} + {(- y)}^{4}$

Step 3.23

Apply the product rule to $- y$ .

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

Step 3.24

Raise $- 1$ to the power of $4$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} - 2 x y^{3} + 1 y^{4}$

Step 3.25

Multiply $y^{4}$ by $1$ .

$\frac{x^{4}}{16} - \frac{y x^{3}}{2} + \frac{3 x^{2} y^{2}}{2} - 2 x y^{3} + y^{4}$