Expand Using Pascal's Triangle (5x-2y)^3

${(5 x - 2 y)}^{3}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

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 ${(5 x - 2 y)}^{3}$ , $n = 3$ so the coefficients of the expansion will correspond with line $4$ .

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 $1 - 3 - 3 - 1$ .

$1 a^{3} b^{0} + 3 a^{2} b + 3 a b^{2} + 1 a^{0} b^{3}$

Step 3

Substitute the actual values of $a$ $5 x$ and $b$ $- 2 y$ into the expression.

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

Step 4

Simplify each term.

Multiply ${(5 x)}^{3}$ by $1$ .

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

Apply the product rule to $5 x$ .

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

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

$125 x^{3} {(- 2 y)}^{0} + 3 {(5 x)}^{2} {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Apply the product rule to $- 2 y$ .

$125 x^{3} ({(- 2)}^{0} y^{0}) + 3 {(5 x)}^{2} {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Rewrite using the commutative property of multiplication.

$125 \cdot {(- 2)}^{0} x^{3} y^{0} + 3 {(5 x)}^{2} {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Anything raised to $0$ is $1$ .

$125 \cdot 1 x^{3} y^{0} + 3 {(5 x)}^{2} {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Multiply $125$ by $1$ .

$125 x^{3} y^{0} + 3 {(5 x)}^{2} {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Anything raised to $0$ is $1$ .

$125 x^{3} \cdot 1 + 3 {(5 x)}^{2} {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Multiply $125$ by $1$ .

$125 x^{3} + 3 {(5 x)}^{2} {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Apply the product rule to $5 x$ .

$125 x^{3} + 3 (5^{2} x^{2}) {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

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

$125 x^{3} + 3 (25 x^{2}) {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Multiply $25$ by $3$ .

$125 x^{3} + 75 x^{2} {(- 2 y)}^{1} + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Simplify.

$125 x^{3} + 75 x^{2} (- 2 y) + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Rewrite using the commutative property of multiplication.

$125 x^{3} + 75 \cdot - 2 x^{2} y + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Multiply $75$ by $- 2$ .

$125 x^{3} - 150 x^{2} y + 3 {(5 x)}^{1} {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Simplify.

$125 x^{3} - 150 x^{2} y + 3 (5 x) {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Multiply $5$ by $3$ .

$125 x^{3} - 150 x^{2} y + 15 x {(- 2 y)}^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Apply the product rule to $- 2 y$ .

$125 x^{3} - 150 x^{2} y + 15 x ({(- 2)}^{2} y^{2}) + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Rewrite using the commutative property of multiplication.

$125 x^{3} - 150 x^{2} y + 15 \cdot {(- 2)}^{2} x y^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

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

$125 x^{3} - 150 x^{2} y + 15 \cdot 4 x y^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Multiply $15$ by $4$ .

$125 x^{3} - 150 x^{2} y + 60 x y^{2} + 1 {(5 x)}^{0} {(- 2 y)}^{3}$

Multiply ${(5 x)}^{0}$ by $1$ .

$125 x^{3} - 150 x^{2} y + 60 x y^{2} + {(5 x)}^{0} {(- 2 y)}^{3}$

Apply the product rule to $5 x$ .

$125 x^{3} - 150 x^{2} y + 60 x y^{2} + 5^{0} x^{0} {(- 2 y)}^{3}$

Anything raised to $0$ is $1$ .

$125 x^{3} - 150 x^{2} y + 60 x y^{2} + 1 x^{0} {(- 2 y)}^{3}$

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

$125 x^{3} - 150 x^{2} y + 60 x y^{2} + x^{0} {(- 2 y)}^{3}$

Anything raised to $0$ is $1$ .

$125 x^{3} - 150 x^{2} y + 60 x y^{2} + 1 {(- 2 y)}^{3}$

Multiply ${(- 2 y)}^{3}$ by $1$ .

$125 x^{3} - 150 x^{2} y + 60 x y^{2} + {(- 2 y)}^{3}$

Apply the product rule to $- 2 y$ .

$125 x^{3} - 150 x^{2} y + 60 x y^{2} + {(- 2)}^{3} y^{3}$

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

$125 x^{3} - 150 x^{2} y + 60 x y^{2} - 8 y^{3}$

Expand Using Pascal's Triangle (5x-2y)^3

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