Basic Math Examples

${(2 - y)}^{3} {(1 + z b)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

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

$(8 + 3 \cdot 2^{2} (- y) + 3 \cdot 2 {(- y)}^{2} + {(- y)}^{3}) {(1 + z b)}^{4}$

Step 2.2

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

$(8 + 3 \cdot 4 (- y) + 3 \cdot 2 {(- y)}^{2} + {(- y)}^{3}) {(1 + z b)}^{4}$

Step 2.3

Multiply $3$ by $4$ .

$(8 + 12 (- y) + 3 \cdot 2 {(- y)}^{2} + {(- y)}^{3}) {(1 + z b)}^{4}$

Step 2.4

Multiply $- 1$ by $12$ .

$(8 - 12 y + 3 \cdot 2 {(- y)}^{2} + {(- y)}^{3}) {(1 + z b)}^{4}$

Step 2.5

Multiply $3$ by $2$ .

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

Step 2.6

Apply the product rule to $- y$ .

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

Step 2.7

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

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

Step 2.8

Multiply $y^{2}$ by $1$ .

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

Step 2.9

Apply the product rule to $- y$ .

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

Step 2.10

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

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

Step 3

Use the Binomial Theorem.

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

Step 4

Simplify each term.

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Step 4.1

One to any power is one.

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

Step 4.2

One to any power is one.

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

Step 4.3

Multiply $4$ by $1$ .

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

Step 4.4

One to any power is one.

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

Step 4.5

Multiply $6$ by $1$ .

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

Step 4.6

Apply the product rule to $z b$ .

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

Step 4.7

Multiply $4$ by $1$ .

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

Step 4.8

Apply the product rule to $z b$ .

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

Step 4.9

Apply the product rule to $z b$ .

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