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

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

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 ${(2 x - 3 y^{2})}^{3}$ , $n = 3$ 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 $a$ $2 x$ and $b$ $- 3 y^{2}$ into the expression.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $2 x$ .

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

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

$8 x^{3} {(- 3 y^{2})}^{0} + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Apply the product rule to $- 3 y^{2}$ .

$8 x^{3} ({(- 3)}^{0} {(y^{2})}^{0}) + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Rewrite using the commutative property of multiplication.

$8 \cdot {(- 3)}^{0} x^{3} {(y^{2})}^{0} + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Anything raised to $0$ is $1$ .

$8 \cdot 1 x^{3} {(y^{2})}^{0} + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply $8$ by $1$ .

$8 x^{3} {(y^{2})}^{0} + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply the exponents in ${(y^{2})}^{0}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$8 x^{3} y^{2 \cdot 0} + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply $2$ by $0$ .

$8 x^{3} y^{0} + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Anything raised to $0$ is $1$ .

$8 x^{3} \cdot 1 + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply $8$ by $1$ .

$8 x^{3} + 3 {(2 x)}^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Apply the product rule to $2 x$ .

$8 x^{3} + 3 (2^{2} x^{2}) {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

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

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

Multiply $4$ by $3$ .

$8 x^{3} + 12 x^{2} {(- 3 y^{2})}^{1} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Simplify.

$8 x^{3} + 12 x^{2} (- 3 y^{2}) + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Rewrite using the commutative property of multiplication.

$8 x^{3} + 12 \cdot - 3 x^{2} y^{2} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply $12$ by $- 3$ .

$8 x^{3} - 36 x^{2} y^{2} + 3 {(2 x)}^{1} {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Simplify.

$8 x^{3} - 36 x^{2} y^{2} + 3 (2 x) {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply $2$ by $3$ .

$8 x^{3} - 36 x^{2} y^{2} + 6 x {(- 3 y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Apply the product rule to $- 3 y^{2}$ .

$8 x^{3} - 36 x^{2} y^{2} + 6 x ({(- 3)}^{2} {(y^{2})}^{2}) + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Rewrite using the commutative property of multiplication.

$8 x^{3} - 36 x^{2} y^{2} + 6 \cdot {(- 3)}^{2} x {(y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

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

$8 x^{3} - 36 x^{2} y^{2} + 6 \cdot 9 x {(y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply $6$ by $9$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x {(y^{2})}^{2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply the exponents in ${(y^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{2 \cdot 2} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Multiply $2$ by $2$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} + 1 {(2 x)}^{0} {(- 3 y^{2})}^{3}$

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

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} + {(2 x)}^{0} {(- 3 y^{2})}^{3}$

Apply the product rule to $2 x$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} + 2^{0} x^{0} {(- 3 y^{2})}^{3}$

Anything raised to $0$ is $1$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} + 1 x^{0} {(- 3 y^{2})}^{3}$

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

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} + x^{0} {(- 3 y^{2})}^{3}$

Anything raised to $0$ is $1$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} + 1 {(- 3 y^{2})}^{3}$

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

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} + {(- 3 y^{2})}^{3}$

Apply the product rule to $- 3 y^{2}$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} + {(- 3)}^{3} {(y^{2})}^{3}$

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

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} - 27 {(y^{2})}^{3}$

Multiply the exponents in ${(y^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} - 27 y^{2 \cdot 3}$

Multiply $2$ by $3$ .

$8 x^{3} - 36 x^{2} y^{2} + 54 x y^{4} - 27 y^{6}$

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

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