Expand Using Pascal's Triangle (3x+4y)^6

${(3 x + 4 y)}^{6}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

$1 - 6 - 15 - 20 - 15 - 6 - 1$

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 ${(3 x + 4 y)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

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 $1 - 6 - 15 - 20 - 15 - 6 - 1$ .

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

Substitute the actual values of $a$ $3 x$ and $b$ $4 y$ into the expression.

$1 {(3 x)}^{6} {(4 y)}^{0} + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Step 4

Simplify each term.

Multiply ${(3 x)}^{6}$ by $1$ .

${(3 x)}^{6} {(4 y)}^{0} + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $3 x$ .

$3^{6} x^{6} {(4 y)}^{0} + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Raise $3$ to the power of $6$ .

$729 x^{6} {(4 y)}^{0} + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $4 y$ .

$729 x^{6} (4^{0} y^{0}) + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Rewrite using the commutative property of multiplication.

$729 \cdot 4^{0} x^{6} y^{0} + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Anything raised to $0$ is $1$ .

$729 \cdot 1 x^{6} y^{0} + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $729$ by $1$ .

$729 x^{6} y^{0} + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Anything raised to $0$ is $1$ .

$729 x^{6} \cdot 1 + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $729$ by $1$ .

$729 x^{6} + 6 {(3 x)}^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $3 x$ .

$729 x^{6} + 6 (3^{5} x^{5}) {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

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

$729 x^{6} + 6 (243 x^{5}) {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $243$ by $6$ .

$729 x^{6} + 1458 x^{5} {(4 y)}^{1} + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Simplify.

$729 x^{6} + 1458 x^{5} (4 y) + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Rewrite using the commutative property of multiplication.

$729 x^{6} + 1458 \cdot 4 x^{5} y + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $1458$ by $4$ .

$729 x^{6} + 5832 x^{5} y + 15 {(3 x)}^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $3 x$ .

$729 x^{6} + 5832 x^{5} y + 15 (3^{4} x^{4}) {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Raise $3$ to the power of $4$ .

$729 x^{6} + 5832 x^{5} y + 15 (81 x^{4}) {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $81$ by $15$ .

$729 x^{6} + 5832 x^{5} y + 1215 x^{4} {(4 y)}^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $4 y$ .

$729 x^{6} + 5832 x^{5} y + 1215 x^{4} (4^{2} y^{2}) + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Rewrite using the commutative property of multiplication.

$729 x^{6} + 5832 x^{5} y + 1215 \cdot 4^{2} x^{4} y^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

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

$729 x^{6} + 5832 x^{5} y + 1215 \cdot 16 x^{4} y^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $1215$ by $16$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 20 {(3 x)}^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $3 x$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 20 (3^{3} x^{3}) {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

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

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 20 (27 x^{3}) {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $27$ by $20$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 540 x^{3} {(4 y)}^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $4 y$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 540 x^{3} (4^{3} y^{3}) + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Rewrite using the commutative property of multiplication.

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 540 \cdot 4^{3} x^{3} y^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Raise $4$ to the power of $3$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 540 \cdot 64 x^{3} y^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $540$ by $64$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 15 {(3 x)}^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $3 x$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 15 (3^{2} x^{2}) {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

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

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 15 (9 x^{2}) {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $9$ by $15$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 135 x^{2} {(4 y)}^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $4 y$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 135 x^{2} (4^{4} y^{4}) + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Rewrite using the commutative property of multiplication.

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 135 \cdot 4^{4} x^{2} y^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

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

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 135 \cdot 256 x^{2} y^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $135$ by $256$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 6 {(3 x)}^{1} {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Simplify.

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 6 (3 x) {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $3$ by $6$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18 x {(4 y)}^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $4 y$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18 x (4^{5} y^{5}) + 1 {(3 x)}^{0} {(4 y)}^{6}$

Rewrite using the commutative property of multiplication.

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18 \cdot 4^{5} x y^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

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

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18 \cdot 1024 x y^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply $18$ by $1024$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + 1 {(3 x)}^{0} {(4 y)}^{6}$

Multiply ${(3 x)}^{0}$ by $1$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + {(3 x)}^{0} {(4 y)}^{6}$

Apply the product rule to $3 x$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + 3^{0} x^{0} {(4 y)}^{6}$

Anything raised to $0$ is $1$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + 1 x^{0} {(4 y)}^{6}$

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

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + x^{0} {(4 y)}^{6}$

Anything raised to $0$ is $1$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + 1 {(4 y)}^{6}$

Multiply ${(4 y)}^{6}$ by $1$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + {(4 y)}^{6}$

Apply the product rule to $4 y$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + 4^{6} y^{6}$

Raise $4$ to the power of $6$ .

$729 x^{6} + 5832 x^{5} y + 19440 x^{4} y^{2} + 34560 x^{3} y^{3} + 34560 x^{2} y^{4} + 18432 x y^{5} + 4096 y^{6}$

Expand Using Pascal's Triangle (3x+4y)^6

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