Expand Using Pascal's Triangle (2x-5y)^5

${(2 x - 5 y)}^{5}$

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 ${(2 x - 5 y)}^{5}$ , $n = 5$ 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 $a$ $2 x$ and $b$ $- 5 y$ into the expression.

$1 {(2 x)}^{5} {(- 5 y)}^{0} + 5 {(2 x)}^{4} {(- 5 y)}^{1} + 10 {(2 x)}^{3} {(- 5 y)}^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Step 4

Simplify each term.

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

${(2 x)}^{5} {(- 5 y)}^{0} + 5 {(2 x)}^{4} {(- 5 y)}^{1} + 10 {(2 x)}^{3} {(- 5 y)}^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Apply the product rule to $2 x$ .

$2^{5} x^{5} {(- 5 y)}^{0} + 5 {(2 x)}^{4} {(- 5 y)}^{1} + 10 {(2 x)}^{3} {(- 5 y)}^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

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

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

Apply the product rule to $- 5 y$ .

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

Rewrite using the commutative property of multiplication.

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

Anything raised to $0$ is $1$ .

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

Multiply $32$ by $1$ .

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

Anything raised to $0$ is $1$ .

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

Multiply $32$ by $1$ .

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

Apply the product rule to $2 x$ .

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

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

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

Multiply $16$ by $5$ .

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Multiply $80$ by $- 5$ .

$32 x^{5} - 400 x^{4} y + 10 {(2 x)}^{3} {(- 5 y)}^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} - 400 x^{4} y + 10 (2^{3} x^{3}) {(- 5 y)}^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

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

$32 x^{5} - 400 x^{4} y + 10 (8 x^{3}) {(- 5 y)}^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Multiply $8$ by $10$ .

$32 x^{5} - 400 x^{4} y + 80 x^{3} {(- 5 y)}^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Apply the product rule to $- 5 y$ .

$32 x^{5} - 400 x^{4} y + 80 x^{3} ({(- 5)}^{2} y^{2}) + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 400 x^{4} y + 80 \cdot {(- 5)}^{2} x^{3} y^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

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

$32 x^{5} - 400 x^{4} y + 80 \cdot 25 x^{3} y^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Multiply $80$ by $25$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} + 10 {(2 x)}^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} + 10 (2^{2} x^{2}) {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

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

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} + 10 (4 x^{2}) {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Multiply $4$ by $10$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} + 40 x^{2} {(- 5 y)}^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Apply the product rule to $- 5 y$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} + 40 x^{2} ({(- 5)}^{3} y^{3}) + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} + 40 \cdot {(- 5)}^{3} x^{2} y^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

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

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} + 40 \cdot - 125 x^{2} y^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Multiply $40$ by $- 125$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 5 {(2 x)}^{1} {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Simplify.

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 5 (2 x) {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Multiply $2$ by $5$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 10 x {(- 5 y)}^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Apply the product rule to $- 5 y$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 10 x ({(- 5)}^{4} y^{4}) + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Rewrite using the commutative property of multiplication.

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 10 \cdot {(- 5)}^{4} x y^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Raise $- 5$ to the power of $4$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 10 \cdot 625 x y^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

Multiply $10$ by $625$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} + 1 {(2 x)}^{0} {(- 5 y)}^{5}$

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

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} + {(2 x)}^{0} {(- 5 y)}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} + 2^{0} x^{0} {(- 5 y)}^{5}$

Anything raised to $0$ is $1$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} + 1 x^{0} {(- 5 y)}^{5}$

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

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} + x^{0} {(- 5 y)}^{5}$

Anything raised to $0$ is $1$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} + 1 {(- 5 y)}^{5}$

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

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} + {(- 5 y)}^{5}$

Apply the product rule to $- 5 y$ .

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} + {(- 5)}^{5} y^{5}$

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

$32 x^{5} - 400 x^{4} y + 2000 x^{3} y^{2} - 5000 x^{2} y^{3} + 6250 x y^{4} - 3125 y^{5}$

Expand Using Pascal's Triangle (2x-5y)^5

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