Expand Using Pascal's Triangle (2a-b)^5

${(2 a - b)}^{5}$

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 ${(2 a - b)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

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

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

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

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

Simplify each term.

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

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

Apply the product rule to $2 a$ .

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

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

$32 a^{5} {(- b)}^{0} + 5 {(2 a)}^{4} {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Apply the product rule to $- b$ .

$32 a^{5} ({(- 1)}^{0} b^{0}) + 5 {(2 a)}^{4} {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Rewrite using the commutative property of multiplication.

$32 \cdot {(- 1)}^{0} a^{5} b^{0} + 5 {(2 a)}^{4} {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Anything raised to $0$ is $1$ .

$32 \cdot 1 a^{5} b^{0} + 5 {(2 a)}^{4} {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $32$ by $1$ .

$32 a^{5} b^{0} + 5 {(2 a)}^{4} {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Anything raised to $0$ is $1$ .

$32 a^{5} \cdot 1 + 5 {(2 a)}^{4} {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $32$ by $1$ .

$32 a^{5} + 5 {(2 a)}^{4} {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Apply the product rule to $2 a$ .

$32 a^{5} + 5 (2^{4} a^{4}) {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

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

$32 a^{5} + 5 (16 a^{4}) {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $16$ by $5$ .

$32 a^{5} + 80 a^{4} {(- b)}^{1} + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Simplify.

$32 a^{5} + 80 a^{4} (- b) + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Rewrite using the commutative property of multiplication.

$32 a^{5} + 80 \cdot - 1 a^{4} b + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $80$ by $- 1$ .

$32 a^{5} - 80 a^{4} b + 10 {(2 a)}^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Apply the product rule to $2 a$ .

$32 a^{5} - 80 a^{4} b + 10 (2^{3} a^{3}) {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

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

$32 a^{5} - 80 a^{4} b + 10 (8 a^{3}) {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $8$ by $10$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} {(- b)}^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Apply the product rule to $- b$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} ({(- 1)}^{2} b^{2}) + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Rewrite using the commutative property of multiplication.

$32 a^{5} - 80 a^{4} b + 80 \cdot {(- 1)}^{2} a^{3} b^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

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

$32 a^{5} - 80 a^{4} b + 80 \cdot 1 a^{3} b^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $80$ by $1$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} + 10 {(2 a)}^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Apply the product rule to $2 a$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} + 10 (2^{2} a^{2}) {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

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

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} + 10 (4 a^{2}) {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $4$ by $10$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} + 40 a^{2} {(- b)}^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Apply the product rule to $- b$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} + 40 a^{2} ({(- 1)}^{3} b^{3}) + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Rewrite using the commutative property of multiplication.

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} + 40 \cdot {(- 1)}^{3} a^{2} b^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

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

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} + 40 \cdot - 1 a^{2} b^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $40$ by $- 1$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 5 {(2 a)}^{1} {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Simplify.

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 5 (2 a) {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $2$ by $5$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a {(- b)}^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Apply the product rule to $- b$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a ({(- 1)}^{4} b^{4}) + 1 {(2 a)}^{0} {(- b)}^{5}$

Rewrite using the commutative property of multiplication.

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 \cdot {(- 1)}^{4} a b^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

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

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 \cdot 1 a b^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

Multiply $10$ by $1$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} + 1 {(2 a)}^{0} {(- b)}^{5}$

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

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} + {(2 a)}^{0} {(- b)}^{5}$

Apply the product rule to $2 a$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} + 2^{0} a^{0} {(- b)}^{5}$

Anything raised to $0$ is $1$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} + 1 a^{0} {(- b)}^{5}$

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

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} + a^{0} {(- b)}^{5}$

Anything raised to $0$ is $1$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} + 1 {(- b)}^{5}$

Multiply ${(- b)}^{5}$ by $1$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} + {(- b)}^{5}$

Apply the product rule to $- b$ .

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} + {(- 1)}^{5} b^{5}$

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

$32 a^{5} - 80 a^{4} b + 80 a^{3} b^{2} - 40 a^{2} b^{3} + 10 a b^{4} - b^{5}$

Expand Using Pascal's Triangle (2a-b)^5

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