Expand Using Pascal's Triangle (m+2)^4

${(m + 2)}^{4}$

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 ${(m + 2)}^{4}$ , $n = 4$ 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 $a$ $m$ and $b$ $2$ into the expression.

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

Step 4

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

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

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

Evaluate the exponent.

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

Multiply $2$ by $4$ .

$m^{4} + 8 m^{3} + 6 {(m)}^{2} {(2)}^{2} + 4 {(m)}^{1} {(2)}^{3} + 1 {(m)}^{0} {(2)}^{4}$

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

$m^{4} + 8 m^{3} + 6 m^{2} \cdot 4 + 4 {(m)}^{1} {(2)}^{3} + 1 {(m)}^{0} {(2)}^{4}$

Multiply $4$ by $6$ .

$m^{4} + 8 m^{3} + 24 m^{2} + 4 {(m)}^{1} {(2)}^{3} + 1 {(m)}^{0} {(2)}^{4}$

Simplify.

$m^{4} + 8 m^{3} + 24 m^{2} + 4 m {(2)}^{3} + 1 {(m)}^{0} {(2)}^{4}$

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

$m^{4} + 8 m^{3} + 24 m^{2} + 4 m \cdot 8 + 1 {(m)}^{0} {(2)}^{4}$

Multiply $8$ by $4$ .

$m^{4} + 8 m^{3} + 24 m^{2} + 32 m + 1 {(m)}^{0} {(2)}^{4}$

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

$m^{4} + 8 m^{3} + 24 m^{2} + 32 m + {(m)}^{0} {(2)}^{4}$

Anything raised to $0$ is $1$ .

$m^{4} + 8 m^{3} + 24 m^{2} + 32 m + 1 {(2)}^{4}$

Multiply ${(2)}^{4}$ by $1$ .

$m^{4} + 8 m^{3} + 24 m^{2} + 32 m + {(2)}^{4}$

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

$m^{4} + 8 m^{3} + 24 m^{2} + 32 m + 16$

Expand Using Pascal's Triangle (m+2)^4

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