Precalculus Examples

${(\frac{1}{2} x - 4 y)}^{5}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(\frac{1}{2} x - 4 y)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 5 - 10 - 10 - 5 - 1$ .

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

Substitute the actual values of $a$ $\frac{1}{2} x$ and $b$ $- 4 y$ into the expression.

$1 {(\frac{1}{2} x)}^{5} {(- 4 y)}^{0} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(\frac{1}{2} x)}^{5}$ by $1$ .

${(\frac{1}{2} x)}^{5} {(- 4 y)}^{0} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.2

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

${(\frac{x}{2})}^{5} {(- 4 y)}^{0} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.3

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

$\frac{x^{5}}{2^{5}} {(- 4 y)}^{0} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.4

Raise $2$ to the power of $5$ .

$\frac{x^{5}}{32} {(- 4 y)}^{0} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.5

Apply the product rule to $- 4 y$ .

$\frac{x^{5}}{32} ({(- 4)}^{0} y^{0}) + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.6

Rewrite using the commutative property of multiplication.

${(- 4)}^{0} \frac{x^{5}}{32} y^{0} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.7

Anything raised to $0$ is $1$ .

$1 \frac{x^{5}}{32} y^{0} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.8

Multiply $\frac{x^{5}}{32}$ by $1$ .

$\frac{x^{5}}{32} y^{0} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.9

Anything raised to $0$ is $1$ .

$\frac{x^{5}}{32} \cdot 1 + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.10

Multiply $\frac{x^{5}}{32}$ by $1$ .

$\frac{x^{5}}{32} + 5 {(\frac{1}{2} x)}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.11

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

$\frac{x^{5}}{32} + 5 {(\frac{x}{2})}^{4} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.12

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

$\frac{x^{5}}{32} + 5 \frac{x^{4}}{2^{4}} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.13

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

$\frac{x^{5}}{32} + 5 \frac{x^{4}}{16} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.14

Combine $5$ and $\frac{x^{4}}{16}$ .

$\frac{x^{5}}{32} + \frac{5 x^{4}}{16} {(- 4 y)}^{1} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.15

Simplify.

$\frac{x^{5}}{32} + \frac{5 x^{4}}{16} (- 4 y) + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.16

Rewrite using the commutative property of multiplication.

$\frac{x^{5}}{32} - 4 \frac{5 x^{4}}{16} y + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.17

Cancel the common factor of $4$ .

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Step 4.17.1

Factor $4$ out of $- 4$ .

$\frac{x^{5}}{32} + 4 (- 1) \frac{5 x^{4}}{16} y + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.17.2

Factor $4$ out of $16$ .

$\frac{x^{5}}{32} + 4 \cdot - 1 \frac{5 x^{4}}{4 \cdot 4} y + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.17.3

Cancel the common factor.

Step 4.17.4

Rewrite the expression.

$\frac{x^{5}}{32} - \frac{5 x^{4}}{4} y + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.18

Combine $y$ and $\frac{5 x^{4}}{4}$ .

$\frac{x^{5}}{32} - \frac{y (5 x^{4})}{4} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.19

Move $5$ to the left of $y$ .

$\frac{x^{5}}{32} - \frac{5 \cdot y x^{4}}{4} + 10 {(\frac{1}{2} x)}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.20

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 10 {(\frac{x}{2})}^{3} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.21

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 10 \frac{x^{3}}{2^{3}} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.22

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 10 \frac{x^{3}}{8} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.23

Cancel the common factor of $2$ .

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Step 4.23.1

Factor $2$ out of $10$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 2 (5) \frac{x^{3}}{8} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.23.2

Factor $2$ out of $8$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 2 \cdot 5 \frac{x^{3}}{2 \cdot 4} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.23.3

Cancel the common factor.

Step 4.23.4

Rewrite the expression.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 5 \frac{x^{3}}{4} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.24

Combine $5$ and $\frac{x^{3}}{4}$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + \frac{5 x^{3}}{4} {(- 4 y)}^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.25

Apply the product rule to $- 4 y$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + \frac{5 x^{3}}{4} ({(- 4)}^{2} y^{2}) + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.26

Rewrite using the commutative property of multiplication.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + {(- 4)}^{2} \frac{5 x^{3}}{4} y^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.27

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 16 \frac{5 x^{3}}{4} y^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.28

Cancel the common factor of $4$ .

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Step 4.28.1

Factor $4$ out of $16$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 4 (4) \frac{5 x^{3}}{4} y^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.28.2

Cancel the common factor.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 4 \cdot 4 \frac{5 x^{3}}{4} y^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.28.3

Rewrite the expression.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 4 (5 x^{3}) y^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.29

Multiply $5$ by $4$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 10 {(\frac{1}{2} x)}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.30

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 10 {(\frac{x}{2})}^{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.31

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 10 \frac{x^{2}}{2^{2}} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.32

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 10 \frac{x^{2}}{4} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.33

Cancel the common factor of $2$ .

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Step 4.33.1

Factor $2$ out of $10$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 2 (5) \frac{x^{2}}{4} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.33.2

Factor $2$ out of $4$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 2 \cdot 5 \frac{x^{2}}{2 \cdot 2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.33.3

Cancel the common factor.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 2 \cdot 5 \frac{x^{2}}{2 \cdot 2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.33.4

Rewrite the expression.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 5 \frac{x^{2}}{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.34

Combine $5$ and $\frac{x^{2}}{2}$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + \frac{5 x^{2}}{2} {(- 4 y)}^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.35

Apply the product rule to $- 4 y$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + \frac{5 x^{2}}{2} ({(- 4)}^{3} y^{3}) + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.36

Rewrite using the commutative property of multiplication.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + {(- 4)}^{3} \frac{5 x^{2}}{2} y^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.37

Raise $- 4$ to the power of $3$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 64 \frac{5 x^{2}}{2} y^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.38

Cancel the common factor of $2$ .

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Step 4.38.1

Factor $2$ out of $- 64$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 2 (- 32) \frac{5 x^{2}}{2} y^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.38.2

Cancel the common factor.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} + 2 \cdot - 32 \frac{5 x^{2}}{2} y^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.38.3

Rewrite the expression.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 32 (5 x^{2}) y^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.39

Multiply $5$ by $- 32$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 5 {(\frac{1}{2} x)}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.40

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 5 {(\frac{x}{2})}^{1} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.41

Simplify.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 5 \frac{x}{2} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.42

Combine $5$ and $\frac{x}{2}$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + \frac{5 x}{2} {(- 4 y)}^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.43

Apply the product rule to $- 4 y$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + \frac{5 x}{2} ({(- 4)}^{4} y^{4}) + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.44

Rewrite using the commutative property of multiplication.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + {(- 4)}^{4} \frac{5 x}{2} y^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.45

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 256 \frac{5 x}{2} y^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.46

Cancel the common factor of $2$ .

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Step 4.46.1

Factor $2$ out of $256$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 2 (128) \frac{5 x}{2} y^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.46.2

Cancel the common factor.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 2 \cdot 128 \frac{5 x}{2} y^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.46.3

Rewrite the expression.

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 128 (5 x) y^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.47

Multiply $5$ by $128$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + 1 {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.48

Multiply ${(\frac{1}{2} x)}^{0}$ by $1$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + {(\frac{1}{2} x)}^{0} {(- 4 y)}^{5}$

Step 4.49

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + {(\frac{x}{2})}^{0} {(- 4 y)}^{5}$

Step 4.50

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

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + \frac{x^{0}}{2^{0}} {(- 4 y)}^{5}$

Step 4.51

Anything raised to $0$ is $1$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + \frac{1}{2^{0}} {(- 4 y)}^{5}$

Step 4.52

Anything raised to $0$ is $1$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + \frac{1}{1} {(- 4 y)}^{5}$

Step 4.53

Divide $1$ by $1$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + 1 {(- 4 y)}^{5}$

Step 4.54

Multiply ${(- 4 y)}^{5}$ by $1$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + {(- 4 y)}^{5}$

Step 4.55

Apply the product rule to $- 4 y$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} + {(- 4)}^{5} y^{5}$

Step 4.56

Raise $- 4$ to the power of $5$ .

$\frac{x^{5}}{32} - \frac{5 y x^{4}}{4} + 20 x^{3} y^{2} - 160 x^{2} y^{3} + 640 x y^{4} - 1024 y^{5}$