Expand Using the Binomial Theorem (6m+n)^2

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(6 m)}^{2 - k} \cdot {(n)}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(6 m)}^{2 - 0} \cdot {(n)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(6 m)}^{2 - 1} \cdot {(n)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(6 m)}^{2 - 2} \cdot {(n)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $6 m$ .

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

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

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

Anything raised to $0$ is $1$ .

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

Multiply $36$ by $1$ .

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

Simplify.

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

Multiply $6$ by $2$ .

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

Simplify.

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

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

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

Apply the product rule to $6 m$ .

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

Anything raised to $0$ is $1$ .

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

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

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

Anything raised to $0$ is $1$ .

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

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

$36 m^{2} + 12 m n + n^{2}$

Expand Using the Binomial Theorem (6m+n)^2

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