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

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

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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 ${(x + 6 y)}^{4}$ , $n = 4$ so the coefficients of the expansion will correspond with line $5$ .

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 $1 - 4 - 6 - 4 - 1$ .

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

Step 3

Substitute the actual values of $a$ $x$ and $b$ $6 y$ into the expression.

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

Step 4

Simplify each term.

Multiply ${(x)}^{4}$ by $1$ .

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

Apply the product rule to $6 y$ .

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

Rewrite using the commutative property of multiplication.

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{4}$ by $1$ .

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

Anything raised to $0$ is $1$ .

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

Multiply $x^{4}$ by $1$ .

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Multiply $4$ by $6$ .

$x^{4} + 24 x^{3} y + 6 {(x)}^{2} {(6 y)}^{2} + 4 {(x)}^{1} {(6 y)}^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

Apply the product rule to $6 y$ .

$x^{4} + 24 x^{3} y + 6 x^{2} (6^{2} y^{2}) + 4 {(x)}^{1} {(6 y)}^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

Rewrite using the commutative property of multiplication.

$x^{4} + 24 x^{3} y + 6 \cdot 6^{2} x^{2} y^{2} + 4 {(x)}^{1} {(6 y)}^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

Multiply $6$ by $6^{2}$ by adding the exponents.

Multiply $6$ by $6^{2}$ .

Raise $6$ to the power of $1$ .

$x^{4} + 24 x^{3} y + 6^{1} \cdot 6^{2} x^{2} y^{2} + 4 {(x)}^{1} {(6 y)}^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + 24 x^{3} y + 6^{1 + 2} x^{2} y^{2} + 4 {(x)}^{1} {(6 y)}^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

Add $1$ and $2$ .

$x^{4} + 24 x^{3} y + 6^{3} x^{2} y^{2} + 4 {(x)}^{1} {(6 y)}^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

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

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 4 {(x)}^{1} {(6 y)}^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

Simplify.

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 4 x {(6 y)}^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

Apply the product rule to $6 y$ .

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 4 x (6^{3} y^{3}) + 1 {(x)}^{0} {(6 y)}^{4}$

Rewrite using the commutative property of multiplication.

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 4 \cdot 6^{3} x y^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

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

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 4 \cdot 216 x y^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

Multiply $4$ by $216$ .

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 864 x y^{3} + 1 {(x)}^{0} {(6 y)}^{4}$

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

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 864 x y^{3} + {(x)}^{0} {(6 y)}^{4}$

Anything raised to $0$ is $1$ .

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 864 x y^{3} + 1 {(6 y)}^{4}$

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

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 864 x y^{3} + {(6 y)}^{4}$

Apply the product rule to $6 y$ .

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 864 x y^{3} + 6^{4} y^{4}$

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

$x^{4} + 24 x^{3} y + 216 x^{2} y^{2} + 864 x y^{3} + 1296 y^{4}$

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

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