Basic Math Examples

${(4 v + 5)}^{7}$

Step 1

Use the Binomial Theorem.

${(4 v)}^{7} + 7 {(4 v)}^{6} \cdot 5 + 21 {(4 v)}^{5} \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $4 v$ .

$4^{7} v^{7} + 7 {(4 v)}^{6} \cdot 5 + 21 {(4 v)}^{5} \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.2

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

$16384 v^{7} + 7 {(4 v)}^{6} \cdot 5 + 21 {(4 v)}^{5} \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.3

Apply the product rule to $4 v$ .

$16384 v^{7} + 7 (4^{6} v^{6}) \cdot 5 + 21 {(4 v)}^{5} \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.4

Raise $4$ to the power of $6$ .

$16384 v^{7} + 7 (4096 v^{6}) \cdot 5 + 21 {(4 v)}^{5} \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.5

Multiply $4096$ by $7$ .

$16384 v^{7} + 28672 v^{6} \cdot 5 + 21 {(4 v)}^{5} \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.6

Multiply $5$ by $28672$ .

$16384 v^{7} + 143360 v^{6} + 21 {(4 v)}^{5} \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.7

Apply the product rule to $4 v$ .

$16384 v^{7} + 143360 v^{6} + 21 (4^{5} v^{5}) \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.8

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

$16384 v^{7} + 143360 v^{6} + 21 (1024 v^{5}) \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.9

Multiply $1024$ by $21$ .

$16384 v^{7} + 143360 v^{6} + 21504 v^{5} \cdot 5^{2} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.10

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

$16384 v^{7} + 143360 v^{6} + 21504 v^{5} \cdot 25 + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.11

Multiply $25$ by $21504$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 35 {(4 v)}^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.12

Apply the product rule to $4 v$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 35 (4^{4} v^{4}) \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.13

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

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 35 (256 v^{4}) \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.14

Multiply $256$ by $35$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 8960 v^{4} \cdot 5^{3} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.15

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

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 8960 v^{4} \cdot 125 + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.16

Multiply $125$ by $8960$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 35 {(4 v)}^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.17

Apply the product rule to $4 v$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 35 (4^{3} v^{3}) \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.18

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

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 35 (64 v^{3}) \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.19

Multiply $64$ by $35$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 2240 v^{3} \cdot 5^{4} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.20

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

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 2240 v^{3} \cdot 625 + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.21

Multiply $625$ by $2240$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 21 {(4 v)}^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.22

Apply the product rule to $4 v$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 21 (4^{2} v^{2}) \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.23

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

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 21 (16 v^{2}) \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.24

Multiply $16$ by $21$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 336 v^{2} \cdot 5^{5} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.25

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

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 336 v^{2} \cdot 3125 + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.26

Multiply $3125$ by $336$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 1050000 v^{2} + 7 (4 v) \cdot 5^{6} + 5^{7}$

Step 2.27

Multiply $4$ by $7$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 1050000 v^{2} + 28 v \cdot 5^{6} + 5^{7}$

Step 2.28

Raise $5$ to the power of $6$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 1050000 v^{2} + 28 v \cdot 15625 + 5^{7}$

Step 2.29

Multiply $15625$ by $28$ .

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 1050000 v^{2} + 437500 v + 5^{7}$

Step 2.30

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

$16384 v^{7} + 143360 v^{6} + 537600 v^{5} + 1120000 v^{4} + 1400000 v^{3} + 1050000 v^{2} + 437500 v + 78125$