Expand Using Pascal's Triangle (3u-v)^6

${(3 u - v)}^{6}$

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 ${(3 u - v)}^{6}$ , $n = 6$ 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 $a$ $3 u$ and $b$ $- v$ into the expression.

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

Step 4

Simplify each term.

Multiply ${(3 u)}^{6}$ by $1$ .

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

Apply the product rule to $3 u$ .

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

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

$729 u^{6} {(- v)}^{0} + 6 {(3 u)}^{5} {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} ({(- 1)}^{0} v^{0}) + 6 {(3 u)}^{5} {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 \cdot {(- 1)}^{0} u^{6} v^{0} + 6 {(3 u)}^{5} {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Anything raised to $0$ is $1$ .

$729 \cdot 1 u^{6} v^{0} + 6 {(3 u)}^{5} {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $729$ by $1$ .

$729 u^{6} v^{0} + 6 {(3 u)}^{5} {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Anything raised to $0$ is $1$ .

$729 u^{6} \cdot 1 + 6 {(3 u)}^{5} {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $729$ by $1$ .

$729 u^{6} + 6 {(3 u)}^{5} {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} + 6 (3^{5} u^{5}) {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

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

$729 u^{6} + 6 (243 u^{5}) {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $243$ by $6$ .

$729 u^{6} + 1458 u^{5} {(- v)}^{1} + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Simplify.

$729 u^{6} + 1458 u^{5} (- v) + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} + 1458 \cdot - 1 u^{5} v + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $1458$ by $- 1$ .

$729 u^{6} - 1458 u^{5} v + 15 {(3 u)}^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} - 1458 u^{5} v + 15 (3^{4} u^{4}) {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 15 (81 u^{4}) {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $81$ by $15$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} {(- v)}^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} ({(- 1)}^{2} v^{2}) + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} - 1458 u^{5} v + 1215 \cdot {(- 1)}^{2} u^{4} v^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 \cdot 1 u^{4} v^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $1215$ by $1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 20 {(3 u)}^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 20 (3^{3} u^{3}) {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 20 (27 u^{3}) {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $27$ by $20$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 540 u^{3} {(- v)}^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 540 u^{3} ({(- 1)}^{3} v^{3}) + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 540 \cdot {(- 1)}^{3} u^{3} v^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} + 540 \cdot - 1 u^{3} v^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $540$ by $- 1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 15 {(3 u)}^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 15 (3^{2} u^{2}) {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 15 (9 u^{2}) {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $9$ by $15$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} {(- v)}^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} ({(- 1)}^{4} v^{4}) + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 \cdot {(- 1)}^{4} u^{2} v^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 \cdot 1 u^{2} v^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $135$ by $1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 6 {(3 u)}^{1} {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Simplify.

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 6 (3 u) {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $3$ by $6$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 18 u {(- v)}^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 18 u ({(- 1)}^{5} v^{5}) + 1 {(3 u)}^{0} {(- v)}^{6}$

Rewrite using the commutative property of multiplication.

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 18 \cdot {(- 1)}^{5} u v^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} + 18 \cdot - 1 u v^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply $18$ by $- 1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + 1 {(3 u)}^{0} {(- v)}^{6}$

Multiply ${(3 u)}^{0}$ by $1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + {(3 u)}^{0} {(- v)}^{6}$

Apply the product rule to $3 u$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + 3^{0} u^{0} {(- v)}^{6}$

Anything raised to $0$ is $1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + 1 u^{0} {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + u^{0} {(- v)}^{6}$

Anything raised to $0$ is $1$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + 1 {(- v)}^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + {(- 1)}^{6} v^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + 1 v^{6}$

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

$729 u^{6} - 1458 u^{5} v + 1215 u^{4} v^{2} - 540 u^{3} v^{3} + 135 u^{2} v^{4} - 18 u v^{5} + v^{6}$

Expand Using Pascal's Triangle (3u-v)^6

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