Basic Math Examples

${(8 a - 11 b)}^{8}$

Step 1

Use the Binomial Theorem.

${(8 a)}^{8} + 8 {(8 a)}^{7} (- 11 b) + 28 {(8 a)}^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $8 a$ .

$8^{8} a^{8} + 8 {(8 a)}^{7} (- 11 b) + 28 {(8 a)}^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.2

Raise $8$ to the power of $8$ .

$16777216 a^{8} + 8 {(8 a)}^{7} (- 11 b) + 28 {(8 a)}^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.3

Rewrite using the commutative property of multiplication.

$16777216 a^{8} + 8 \cdot - 11 {(8 a)}^{7} b + 28 {(8 a)}^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.4

Multiply $8$ by $- 11$ .

$16777216 a^{8} - 88 {(8 a)}^{7} b + 28 {(8 a)}^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.5

Apply the product rule to $8 a$ .

$16777216 a^{8} - 88 (8^{7} a^{7}) b + 28 {(8 a)}^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.6

Raise $8$ to the power of $7$ .

$16777216 a^{8} - 88 (2097152 a^{7}) b + 28 {(8 a)}^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.7

Multiply $2097152$ by $- 88$ .

$16777216 a^{8} - 184549376 a^{7} b + 28 {(8 a)}^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.8

Apply the product rule to $8 a$ .

$16777216 a^{8} - 184549376 a^{7} b + 28 (8^{6} a^{6}) {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.9

Raise $8$ to the power of $6$ .

$16777216 a^{8} - 184549376 a^{7} b + 28 (262144 a^{6}) {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.10

Multiply $262144$ by $28$ .

$16777216 a^{8} - 184549376 a^{7} b + 7340032 a^{6} {(- 11 b)}^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.11

Apply the product rule to $- 11 b$ .

$16777216 a^{8} - 184549376 a^{7} b + 7340032 a^{6} ({(- 11)}^{2} b^{2}) + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.12

Rewrite using the commutative property of multiplication.

$16777216 a^{8} - 184549376 a^{7} b + 7340032 \cdot {(- 11)}^{2} a^{6} b^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.13

Raise $- 11$ to the power of $2$ .

$16777216 a^{8} - 184549376 a^{7} b + 7340032 \cdot 121 a^{6} b^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.14

Multiply $7340032$ by $121$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} + 56 {(8 a)}^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.15

Apply the product rule to $8 a$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} + 56 (8^{5} a^{5}) {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.16

Raise $8$ to the power of $5$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} + 56 (32768 a^{5}) {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.17

Multiply $32768$ by $56$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} + 1835008 a^{5} {(- 11 b)}^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.18

Apply the product rule to $- 11 b$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} + 1835008 a^{5} ({(- 11)}^{3} b^{3}) + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.19

Rewrite using the commutative property of multiplication.

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} + 1835008 \cdot {(- 11)}^{3} a^{5} b^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.20

Raise $- 11$ to the power of $3$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} + 1835008 \cdot - 1331 a^{5} b^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.21

Multiply $1835008$ by $- 1331$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 70 {(8 a)}^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.22

Apply the product rule to $8 a$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 70 (8^{4} a^{4}) {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.23

Raise $8$ to the power of $4$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 70 (4096 a^{4}) {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.24

Multiply $4096$ by $70$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 286720 a^{4} {(- 11 b)}^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.25

Apply the product rule to $- 11 b$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 286720 a^{4} ({(- 11)}^{4} b^{4}) + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.26

Rewrite using the commutative property of multiplication.

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 286720 \cdot {(- 11)}^{4} a^{4} b^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.27

Raise $- 11$ to the power of $4$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 286720 \cdot 14641 a^{4} b^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.28

Multiply $286720$ by $14641$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} + 56 {(8 a)}^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.29

Apply the product rule to $8 a$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} + 56 (8^{3} a^{3}) {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.30

Raise $8$ to the power of $3$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} + 56 (512 a^{3}) {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.31

Multiply $512$ by $56$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} + 28672 a^{3} {(- 11 b)}^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.32

Apply the product rule to $- 11 b$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} + 28672 a^{3} ({(- 11)}^{5} b^{5}) + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.33

Rewrite using the commutative property of multiplication.

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} + 28672 \cdot {(- 11)}^{5} a^{3} b^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.34

Raise $- 11$ to the power of $5$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} + 28672 \cdot - 161051 a^{3} b^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.35

Multiply $28672$ by $- 161051$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 28 {(8 a)}^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.36

Apply the product rule to $8 a$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 28 (8^{2} a^{2}) {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.37

Raise $8$ to the power of $2$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 28 (64 a^{2}) {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.38

Multiply $64$ by $28$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 1792 a^{2} {(- 11 b)}^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.39

Apply the product rule to $- 11 b$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 1792 a^{2} ({(- 11)}^{6} b^{6}) + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.40

Rewrite using the commutative property of multiplication.

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 1792 \cdot {(- 11)}^{6} a^{2} b^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.41

Raise $- 11$ to the power of $6$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 1792 \cdot 1771561 a^{2} b^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.42

Multiply $1792$ by $1771561$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 3174637312 a^{2} b^{6} + 8 (8 a) {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.43

Multiply $8$ by $8$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 3174637312 a^{2} b^{6} + 64 a {(- 11 b)}^{7} + {(- 11 b)}^{8}$

Step 2.44

Apply the product rule to $- 11 b$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 3174637312 a^{2} b^{6} + 64 a ({(- 11)}^{7} b^{7}) + {(- 11 b)}^{8}$

Step 2.45

Rewrite using the commutative property of multiplication.

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 3174637312 a^{2} b^{6} + 64 \cdot {(- 11)}^{7} a b^{7} + {(- 11 b)}^{8}$

Step 2.46

Raise $- 11$ to the power of $7$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 3174637312 a^{2} b^{6} + 64 \cdot - 19487171 a b^{7} + {(- 11 b)}^{8}$

Step 2.47

Multiply $64$ by $- 19487171$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 3174637312 a^{2} b^{6} - 1247178944 a b^{7} + {(- 11 b)}^{8}$

Step 2.48

Apply the product rule to $- 11 b$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 3174637312 a^{2} b^{6} - 1247178944 a b^{7} + {(- 11)}^{8} b^{8}$

Step 2.49

Raise $- 11$ to the power of $8$ .

$16777216 a^{8} - 184549376 a^{7} b + 888143872 a^{6} b^{2} - 2442395648 a^{5} b^{3} + 4197867520 a^{4} b^{4} - 4617654272 a^{3} b^{5} + 3174637312 a^{2} b^{6} - 1247178944 a b^{7} + 214358881 b^{8}$