Expand Using Pascal's Triangle (6m+2)^2

${(6 m + 2)}^{2}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

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 ${(6 m + 2)}^{2}$ , $n = 2$ so the coefficients of the expansion will correspond with line $3$ .

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 $1 - 2 - 1$ .

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

Step 3

Substitute the actual values of $a$ $6 m$ and $b$ $2$ into the expression.

$1 {(6 m)}^{2} {(2)}^{0} + 2 {(6 m)}^{1} {(2)}^{1} + 1 {(6 m)}^{0} {(2)}^{2}$

Step 4

Simplify each term.

Multiply ${(6 m)}^{2}$ by $1$ .

${(6 m)}^{2} {(2)}^{0} + 2 {(6 m)}^{1} {(2)}^{1} + 1 {(6 m)}^{0} {(2)}^{2}$

Apply the product rule to $6 m$ .

$6^{2} m^{2} {(2)}^{0} + 2 {(6 m)}^{1} {(2)}^{1} + 1 {(6 m)}^{0} {(2)}^{2}$

Raise $6$ to the power of $2$ .

$36 m^{2} {(2)}^{0} + 2 {(6 m)}^{1} {(2)}^{1} + 1 {(6 m)}^{0} {(2)}^{2}$

Anything raised to $0$ is $1$ .

$36 m^{2} \cdot 1 + 2 {(6 m)}^{1} {(2)}^{1} + 1 {(6 m)}^{0} {(2)}^{2}$

Multiply $36$ by $1$ .

$36 m^{2} + 2 {(6 m)}^{1} {(2)}^{1} + 1 {(6 m)}^{0} {(2)}^{2}$

Simplify.

$36 m^{2} + 2 (6 m) {(2)}^{1} + 1 {(6 m)}^{0} {(2)}^{2}$

Multiply $6$ by $2$ .

$36 m^{2} + 12 m {(2)}^{1} + 1 {(6 m)}^{0} {(2)}^{2}$

Evaluate the exponent.

$36 m^{2} + 12 m \cdot 2 + 1 {(6 m)}^{0} {(2)}^{2}$

Multiply $2$ by $12$ .

$36 m^{2} + 24 m + 1 {(6 m)}^{0} {(2)}^{2}$

Multiply ${(6 m)}^{0}$ by $1$ .

$36 m^{2} + 24 m + {(6 m)}^{0} {(2)}^{2}$

Apply the product rule to $6 m$ .

$36 m^{2} + 24 m + 6^{0} m^{0} {(2)}^{2}$

Anything raised to $0$ is $1$ .

$36 m^{2} + 24 m + 1 m^{0} {(2)}^{2}$

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

$36 m^{2} + 24 m + m^{0} {(2)}^{2}$

Anything raised to $0$ is $1$ .

$36 m^{2} + 24 m + 1 {(2)}^{2}$

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

$36 m^{2} + 24 m + {(2)}^{2}$

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

$36 m^{2} + 24 m + 4$

Expand Using Pascal's Triangle (6m+2)^2

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