Basic Math Examples

${(3 y + 7)}^{5} - 4 {(3 y)}^{3}$

Step 1

Use the Binomial Theorem.

${(3 y)}^{5} + 5 {(3 y)}^{4} \cdot 7 + 10 {(3 y)}^{3} \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $3 y$ .

$3^{5} y^{5} + 5 {(3 y)}^{4} \cdot 7 + 10 {(3 y)}^{3} \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.2

Raise $3$ to the power of $5$ .

$243 y^{5} + 5 {(3 y)}^{4} \cdot 7 + 10 {(3 y)}^{3} \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.3

Apply the product rule to $3 y$ .

$243 y^{5} + 5 (3^{4} y^{4}) \cdot 7 + 10 {(3 y)}^{3} \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.4

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

$243 y^{5} + 5 (81 y^{4}) \cdot 7 + 10 {(3 y)}^{3} \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.5

Multiply $81$ by $5$ .

$243 y^{5} + 405 y^{4} \cdot 7 + 10 {(3 y)}^{3} \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.6

Multiply $7$ by $405$ .

$243 y^{5} + 2835 y^{4} + 10 {(3 y)}^{3} \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.7

Apply the product rule to $3 y$ .

$243 y^{5} + 2835 y^{4} + 10 (3^{3} y^{3}) \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.8

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

$243 y^{5} + 2835 y^{4} + 10 (27 y^{3}) \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.9

Multiply $27$ by $10$ .

$243 y^{5} + 2835 y^{4} + 270 y^{3} \cdot 7^{2} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.10

Raise $7$ to the power of $2$ .

$243 y^{5} + 2835 y^{4} + 270 y^{3} \cdot 49 + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.11

Multiply $49$ by $270$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 10 {(3 y)}^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.12

Apply the product rule to $3 y$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 10 (3^{2} y^{2}) \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.13

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

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 10 (9 y^{2}) \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.14

Multiply $9$ by $10$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 90 y^{2} \cdot 7^{3} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.15

Raise $7$ to the power of $3$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 90 y^{2} \cdot 343 + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.16

Multiply $343$ by $90$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 30870 y^{2} + 5 (3 y) \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.17

Multiply $3$ by $5$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 30870 y^{2} + 15 y \cdot 7^{4} + 7^{5} - 4 {(3 y)}^{3}$

Step 2.18

Raise $7$ to the power of $4$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 30870 y^{2} + 15 y \cdot 2401 + 7^{5} - 4 {(3 y)}^{3}$

Step 2.19

Multiply $2401$ by $15$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 30870 y^{2} + 36015 y + 7^{5} - 4 {(3 y)}^{3}$

Step 2.20

Raise $7$ to the power of $5$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 30870 y^{2} + 36015 y + 16807 - 4 {(3 y)}^{3}$

Step 3

Apply the product rule to $3 y$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 30870 y^{2} + 36015 y + 16807 - 4 (3^{3} y^{3})$

Step 4

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

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 30870 y^{2} + 36015 y + 16807 - 4 (27 y^{3})$

Step 5

Multiply $27$ by $- 4$ .

$243 y^{5} + 2835 y^{4} + 13230 y^{3} + 30870 y^{2} + 36015 y + 16807 - 108 y^{3}$

Step 6

Subtract $108 y^{3}$ from $13230 y^{3}$ .

$243 y^{5} + 2835 y^{4} + 13122 y^{3} + 30870 y^{2} + 36015 y + 16807$