Basic Math Examples

${(c - 2 d)}^{6}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

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

Step 2.2

Multiply $6$ by $- 2$ .

$c^{6} - 12 c^{5} d + 15 c^{4} {(- 2 d)}^{2} + 20 c^{3} {(- 2 d)}^{3} + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.3

Apply the product rule to $- 2 d$ .

$c^{6} - 12 c^{5} d + 15 c^{4} ({(- 2)}^{2} d^{2}) + 20 c^{3} {(- 2 d)}^{3} + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.4

Rewrite using the commutative property of multiplication.

$c^{6} - 12 c^{5} d + 15 \cdot {(- 2)}^{2} c^{4} d^{2} + 20 c^{3} {(- 2 d)}^{3} + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.5

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

$c^{6} - 12 c^{5} d + 15 \cdot 4 c^{4} d^{2} + 20 c^{3} {(- 2 d)}^{3} + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.6

Multiply $15$ by $4$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} + 20 c^{3} {(- 2 d)}^{3} + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.7

Apply the product rule to $- 2 d$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} + 20 c^{3} ({(- 2)}^{3} d^{3}) + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.8

Rewrite using the commutative property of multiplication.

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} + 20 \cdot {(- 2)}^{3} c^{3} d^{3} + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.9

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

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} + 20 \cdot - 8 c^{3} d^{3} + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.10

Multiply $20$ by $- 8$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 15 c^{2} {(- 2 d)}^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.11

Apply the product rule to $- 2 d$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 15 c^{2} ({(- 2)}^{4} d^{4}) + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.12

Rewrite using the commutative property of multiplication.

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 15 \cdot {(- 2)}^{4} c^{2} d^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.13

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

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 15 \cdot 16 c^{2} d^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.14

Multiply $15$ by $16$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 240 c^{2} d^{4} + 6 c {(- 2 d)}^{5} + {(- 2 d)}^{6}$

Step 2.15

Apply the product rule to $- 2 d$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 240 c^{2} d^{4} + 6 c ({(- 2)}^{5} d^{5}) + {(- 2 d)}^{6}$

Step 2.16

Rewrite using the commutative property of multiplication.

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 240 c^{2} d^{4} + 6 \cdot {(- 2)}^{5} c d^{5} + {(- 2 d)}^{6}$

Step 2.17

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

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 240 c^{2} d^{4} + 6 \cdot - 32 c d^{5} + {(- 2 d)}^{6}$

Step 2.18

Multiply $6$ by $- 32$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 240 c^{2} d^{4} - 192 c d^{5} + {(- 2 d)}^{6}$

Step 2.19

Apply the product rule to $- 2 d$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 240 c^{2} d^{4} - 192 c d^{5} + {(- 2)}^{6} d^{6}$

Step 2.20

Raise $- 2$ to the power of $6$ .

$c^{6} - 12 c^{5} d + 60 c^{4} d^{2} - 160 c^{3} d^{3} + 240 c^{2} d^{4} - 192 c d^{5} + 64 d^{6}$