Algebra Examples

${(3 c + d^{2})}^{6}$

Step 1

Use the Binomial Theorem.

${(3 c)}^{6} + 6 {(3 c)}^{5} d^{2} + 15 {(3 c)}^{4} {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $3 c$ .

$3^{6} c^{6} + 6 {(3 c)}^{5} d^{2} + 15 {(3 c)}^{4} {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.2

Raise $3$ to the power of $6$ .

$729 c^{6} + 6 {(3 c)}^{5} d^{2} + 15 {(3 c)}^{4} {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.3

Apply the product rule to $3 c$ .

$729 c^{6} + 6 (3^{5} c^{5}) d^{2} + 15 {(3 c)}^{4} {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.4

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

$729 c^{6} + 6 (243 c^{5}) d^{2} + 15 {(3 c)}^{4} {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.5

Multiply $243$ by $6$ .

$729 c^{6} + 1458 c^{5} d^{2} + 15 {(3 c)}^{4} {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.6

Apply the product rule to $3 c$ .

$729 c^{6} + 1458 c^{5} d^{2} + 15 (3^{4} c^{4}) {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.7

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

$729 c^{6} + 1458 c^{5} d^{2} + 15 (81 c^{4}) {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.8

Multiply $81$ by $15$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} {(d^{2})}^{2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.9

Multiply the exponents in ${(d^{2})}^{2}$ .

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Step 2.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{2 \cdot 2} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.9.2

Multiply $2$ by $2$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 20 {(3 c)}^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.10

Apply the product rule to $3 c$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 20 (3^{3} c^{3}) {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.11

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

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 20 (27 c^{3}) {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.12

Multiply $27$ by $20$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} {(d^{2})}^{3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.13

Multiply the exponents in ${(d^{2})}^{3}$ .

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Step 2.13.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{2 \cdot 3} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.13.2

Multiply $2$ by $3$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 15 {(3 c)}^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.14

Apply the product rule to $3 c$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 15 (3^{2} c^{2}) {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.15

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

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 15 (9 c^{2}) {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.16

Multiply $9$ by $15$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 135 c^{2} {(d^{2})}^{4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.17

Multiply the exponents in ${(d^{2})}^{4}$ .

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Step 2.17.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 135 c^{2} d^{2 \cdot 4} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.17.2

Multiply $2$ by $4$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 135 c^{2} d^{8} + 6 (3 c) {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.18

Multiply $3$ by $6$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 135 c^{2} d^{8} + 18 c {(d^{2})}^{5} + {(d^{2})}^{6}$

Step 2.19

Multiply the exponents in ${(d^{2})}^{5}$ .

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Step 2.19.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 135 c^{2} d^{8} + 18 c d^{2 \cdot 5} + {(d^{2})}^{6}$

Step 2.19.2

Multiply $2$ by $5$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 135 c^{2} d^{8} + 18 c d^{10} + {(d^{2})}^{6}$

Step 2.20

Multiply the exponents in ${(d^{2})}^{6}$ .

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Step 2.20.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 135 c^{2} d^{8} + 18 c d^{10} + d^{2 \cdot 6}$

Step 2.20.2

Multiply $2$ by $6$ .

$729 c^{6} + 1458 c^{5} d^{2} + 1215 c^{4} d^{4} + 540 c^{3} d^{6} + 135 c^{2} d^{8} + 18 c d^{10} + d^{12}$