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

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

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 ${(3 + y)}^{4}$ , $n = 4$ 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 $a$ $3$ and $b$ $y$ into the expression.

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

Step 4

Simplify each term.

Multiply ${(3)}^{4}$ by $1$ .

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

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

$81 {(y)}^{0} + 4 {(3)}^{3} {(y)}^{1} + 6 {(3)}^{2} {(y)}^{2} + 4 {(3)}^{1} {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

Anything raised to $0$ is $1$ .

$81 \cdot 1 + 4 {(3)}^{3} {(y)}^{1} + 6 {(3)}^{2} {(y)}^{2} + 4 {(3)}^{1} {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

Multiply $81$ by $1$ .

$81 + 4 {(3)}^{3} {(y)}^{1} + 6 {(3)}^{2} {(y)}^{2} + 4 {(3)}^{1} {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

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

$81 + 4 \cdot 27 {(y)}^{1} + 6 {(3)}^{2} {(y)}^{2} + 4 {(3)}^{1} {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

Multiply $4$ by $27$ .

$81 + 108 {(y)}^{1} + 6 {(3)}^{2} {(y)}^{2} + 4 {(3)}^{1} {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

Simplify.

$81 + 108 y + 6 {(3)}^{2} {(y)}^{2} + 4 {(3)}^{1} {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

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

$81 + 108 y + 6 \cdot 9 {(y)}^{2} + 4 {(3)}^{1} {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

Multiply $6$ by $9$ .

$81 + 108 y + 54 {(y)}^{2} + 4 {(3)}^{1} {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

Evaluate the exponent.

$81 + 108 y + 54 y^{2} + 4 \cdot 3 {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

Multiply $4$ by $3$ .

$81 + 108 y + 54 y^{2} + 12 {(y)}^{3} + 1 {(3)}^{0} {(y)}^{4}$

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

$81 + 108 y + 54 y^{2} + 12 y^{3} + {(3)}^{0} {(y)}^{4}$

Anything raised to $0$ is $1$ .

$81 + 108 y + 54 y^{2} + 12 y^{3} + 1 {(y)}^{4}$

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

$81 + 108 y + 54 y^{2} + 12 y^{3} + y^{4}$

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

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