Expand Using Pascal's Triangle (2u-v)^6

${(2 u - v)}^{6}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(2 u - v)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 6 - 15 - 20 - 15 - 6 - 1$ .

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

Substitute the actual values of $a$ $2 u$ and $b$ $- v$ into the expression.

$1 {(2 u)}^{6} {(- v)}^{0} + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Step 4

Simplify each term.

Multiply ${(2 u)}^{6}$ by $1$ .

${(2 u)}^{6} {(- v)}^{0} + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $2 u$ .

$2^{6} u^{6} {(- v)}^{0} + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

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

$64 u^{6} {(- v)}^{0} + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} ({(- 1)}^{0} v^{0}) + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 \cdot {(- 1)}^{0} u^{6} v^{0} + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Anything raised to $0$ is $1$ .

$64 \cdot 1 u^{6} v^{0} + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $64$ by $1$ .

$64 u^{6} v^{0} + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Anything raised to $0$ is $1$ .

$64 u^{6} \cdot 1 + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $64$ by $1$ .

$64 u^{6} + 6 {(2 u)}^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} + 6 (2^{5} u^{5}) {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

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

$64 u^{6} + 6 (32 u^{5}) {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $32$ by $6$ .

$64 u^{6} + 192 u^{5} {(- v)}^{1} + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Simplify.

$64 u^{6} + 192 u^{5} (- v) + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} + 192 \cdot - 1 u^{5} v + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $192$ by $- 1$ .

$64 u^{6} - 192 u^{5} v + 15 {(2 u)}^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} - 192 u^{5} v + 15 (2^{4} u^{4}) {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

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

$64 u^{6} - 192 u^{5} v + 15 (16 u^{4}) {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $16$ by $15$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} {(- v)}^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} ({(- 1)}^{2} v^{2}) + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} - 192 u^{5} v + 240 \cdot {(- 1)}^{2} u^{4} v^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Raise $- 1$ to the power of $2$ .

$64 u^{6} - 192 u^{5} v + 240 \cdot 1 u^{4} v^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $240$ by $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 20 {(2 u)}^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 20 (2^{3} u^{3}) {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

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

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 20 (8 u^{3}) {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $8$ by $20$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 160 u^{3} {(- v)}^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 160 u^{3} ({(- 1)}^{3} v^{3}) + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 160 \cdot {(- 1)}^{3} u^{3} v^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Raise $- 1$ to the power of $3$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} + 160 \cdot - 1 u^{3} v^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $160$ by $- 1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 15 {(2 u)}^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 15 (2^{2} u^{2}) {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

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

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 15 (4 u^{2}) {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $4$ by $15$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} {(- v)}^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} ({(- 1)}^{4} v^{4}) + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 \cdot {(- 1)}^{4} u^{2} v^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Raise $- 1$ to the power of $4$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 \cdot 1 u^{2} v^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $60$ by $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 6 {(2 u)}^{1} {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Simplify.

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 6 (2 u) {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $2$ by $6$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 12 u {(- v)}^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 12 u ({(- 1)}^{5} v^{5}) + 1 {(2 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 12 \cdot {(- 1)}^{5} u v^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Raise $- 1$ to the power of $5$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} + 12 \cdot - 1 u v^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply $12$ by $- 1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + 1 {(2 u)}^{0} {(- v)}^{6}$

Multiply ${(2 u)}^{0}$ by $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + {(2 u)}^{0} {(- v)}^{6}$

Apply the product rule to $2 u$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + 2^{0} u^{0} {(- v)}^{6}$

Anything raised to $0$ is $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + 1 u^{0} {(- v)}^{6}$

Multiply $u^{0}$ by $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + u^{0} {(- v)}^{6}$

Anything raised to $0$ is $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + 1 {(- v)}^{6}$

Multiply ${(- v)}^{6}$ by $1$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + {(- 1)}^{6} v^{6}$

Raise $- 1$ to the power of $6$ .

$64 u^{6} - 192 u^{5} v + 240 u^{4} v^{2} - 160 u^{3} v^{3} + 60 u^{2} v^{4} - 12 u v^{5} + 1 v^{6}$

Multiply $v^{6}$ by $1$ .

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

Expand Using Pascal's Triangle (2u-v)^6

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