Expand Using Pascal's Triangle (2v+s)^5

${(2 v + s)}^{5}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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 ${(2 v + s)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

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

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

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

$1 {(2 v)}^{5} {(s)}^{0} + 5 {(2 v)}^{4} {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Step 4

Simplify each term.

Multiply ${(2 v)}^{5}$ by $1$ .

${(2 v)}^{5} {(s)}^{0} + 5 {(2 v)}^{4} {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Apply the product rule to $2 v$ .

$2^{5} v^{5} {(s)}^{0} + 5 {(2 v)}^{4} {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

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

$32 v^{5} {(s)}^{0} + 5 {(2 v)}^{4} {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Anything raised to $0$ is $1$ .

$32 v^{5} \cdot 1 + 5 {(2 v)}^{4} {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Multiply $32$ by $1$ .

$32 v^{5} + 5 {(2 v)}^{4} {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Apply the product rule to $2 v$ .

$32 v^{5} + 5 (2^{4} v^{4}) {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

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

$32 v^{5} + 5 (16 v^{4}) {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Multiply $16$ by $5$ .

$32 v^{5} + 80 v^{4} {(s)}^{1} + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Simplify.

$32 v^{5} + 80 v^{4} s + 10 {(2 v)}^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Apply the product rule to $2 v$ .

$32 v^{5} + 80 v^{4} s + 10 (2^{3} v^{3}) {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

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

$32 v^{5} + 80 v^{4} s + 10 (8 v^{3}) {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Multiply $8$ by $10$ .

$32 v^{5} + 80 v^{4} s + 80 v^{3} {(s)}^{2} + 10 {(2 v)}^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Apply the product rule to $2 v$ .

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 10 (2^{2} v^{2}) {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

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

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 10 (4 v^{2}) {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Multiply $4$ by $10$ .

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} {(s)}^{3} + 5 {(2 v)}^{1} {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Simplify.

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} s^{3} + 5 (2 v) {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

Multiply $2$ by $5$ .

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} s^{3} + 10 v {(s)}^{4} + 1 {(2 v)}^{0} {(s)}^{5}$

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

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} s^{3} + 10 v s^{4} + {(2 v)}^{0} {(s)}^{5}$

Apply the product rule to $2 v$ .

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} s^{3} + 10 v s^{4} + 2^{0} v^{0} {(s)}^{5}$

Anything raised to $0$ is $1$ .

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} s^{3} + 10 v s^{4} + 1 v^{0} {(s)}^{5}$

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

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} s^{3} + 10 v s^{4} + v^{0} {(s)}^{5}$

Anything raised to $0$ is $1$ .

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} s^{3} + 10 v s^{4} + 1 {(s)}^{5}$

Multiply ${(s)}^{5}$ by $1$ .

$32 v^{5} + 80 v^{4} s + 80 v^{3} s^{2} + 40 v^{2} s^{3} + 10 v s^{4} + s^{5}$

Expand Using Pascal's Triangle (2v+s)^5

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