Basic Math Examples

${(2 z - y)}^{5}$

Step 1

Use the Binomial Theorem.

${(2 z)}^{5} + 5 {(2 z)}^{4} (- y) + 10 {(2 z)}^{3} {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $2 z$ .

$2^{5} z^{5} + 5 {(2 z)}^{4} (- y) + 10 {(2 z)}^{3} {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.2

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

$32 z^{5} + 5 {(2 z)}^{4} (- y) + 10 {(2 z)}^{3} {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.1.4

Multiply $5$ by $- 1$ .

$32 z^{5} - 5 {(2 z)}^{4} y + 10 {(2 z)}^{3} {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.5

Apply the product rule to $2 z$ .

$32 z^{5} - 5 (2^{4} z^{4}) y + 10 {(2 z)}^{3} {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.6

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

$32 z^{5} - 5 (16 z^{4}) y + 10 {(2 z)}^{3} {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.7

Multiply $16$ by $- 5$ .

$32 z^{5} - 80 z^{4} y + 10 {(2 z)}^{3} {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.8

Apply the product rule to $2 z$ .

$32 z^{5} - 80 z^{4} y + 10 (2^{3} z^{3}) {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.9

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

$32 z^{5} - 80 z^{4} y + 10 (8 z^{3}) {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.10

Multiply $8$ by $10$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} {(- y)}^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.11

Apply the product rule to $- y$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} ({(- 1)}^{2} y^{2}) + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.12

Rewrite using the commutative property of multiplication.

$32 z^{5} - 80 z^{4} y + 80 \cdot {(- 1)}^{2} z^{3} y^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.13

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

$32 z^{5} - 80 z^{4} y + 80 \cdot 1 z^{3} y^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.14

Multiply $80$ by $1$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} + 10 {(2 z)}^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.15

Apply the product rule to $2 z$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} + 10 (2^{2} z^{2}) {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.16

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

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} + 10 (4 z^{2}) {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.17

Multiply $4$ by $10$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} + 40 z^{2} {(- y)}^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.18

Apply the product rule to $- y$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} + 40 z^{2} ({(- 1)}^{3} y^{3}) + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.19

Rewrite using the commutative property of multiplication.

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} + 40 \cdot {(- 1)}^{3} z^{2} y^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.20

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

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} + 40 \cdot - 1 z^{2} y^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.21

Multiply $40$ by $- 1$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 5 (2 z) {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.22

Multiply $2$ by $5$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 10 z {(- y)}^{4} + {(- y)}^{5}$

Step 2.1.23

Apply the product rule to $- y$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 10 z ({(- 1)}^{4} y^{4}) + {(- y)}^{5}$

Step 2.1.24

Rewrite using the commutative property of multiplication.

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 10 \cdot {(- 1)}^{4} z y^{4} + {(- y)}^{5}$

Step 2.1.25

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

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 10 \cdot 1 z y^{4} + {(- y)}^{5}$

Step 2.1.26

Multiply $10$ by $1$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 10 z y^{4} + {(- y)}^{5}$

Step 2.1.27

Apply the product rule to $- y$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 10 z y^{4} + {(- 1)}^{5} y^{5}$

Step 2.1.28

Raise $- 1$ to the power of $5$ .

$32 z^{5} - 80 z^{4} y + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 10 z y^{4} - y^{5}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $z^{4}$ .

$32 z^{5} - 80 y z^{4} + 80 z^{3} y^{2} - 40 z^{2} y^{3} + 10 z y^{4} - y^{5}$

Step 2.2.2

Move $z^{3}$ .

$32 z^{5} - 80 y z^{4} + 80 y^{2} z^{3} - 40 z^{2} y^{3} + 10 z y^{4} - y^{5}$

Step 2.2.3

Move $z^{2}$ .

$32 z^{5} - 80 y z^{4} + 80 y^{2} z^{3} - 40 y^{3} z^{2} + 10 z y^{4} - y^{5}$

Step 2.2.4

Move $z$ .

$32 z^{5} - 80 y z^{4} + 80 y^{2} z^{3} - 40 y^{3} z^{2} + 10 y^{4} z - y^{5}$

Step 2.2.5

Move $32 z^{5}$ .

$- 80 y z^{4} + 80 y^{2} z^{3} - 40 y^{3} z^{2} + 10 y^{4} z - y^{5} + 32 z^{5}$

Step 2.2.6

Move $- 80 y z^{4}$ .

$80 y^{2} z^{3} - 40 y^{3} z^{2} + 10 y^{4} z - y^{5} - 80 y z^{4} + 32 z^{5}$

Step 2.2.7

Move $80 y^{2} z^{3}$ .

$- 40 y^{3} z^{2} + 10 y^{4} z - y^{5} + 80 y^{2} z^{3} - 80 y z^{4} + 32 z^{5}$

Step 2.2.8

Move $- 40 y^{3} z^{2}$ .

$10 y^{4} z - y^{5} - 40 y^{3} z^{2} + 80 y^{2} z^{3} - 80 y z^{4} + 32 z^{5}$

Step 2.2.9

Reorder $10 y^{4} z$ and $- y^{5}$ .

$- y^{5} + 10 y^{4} z - 40 y^{3} z^{2} + 80 y^{2} z^{3} - 80 y z^{4} + 32 z^{5}$