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

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(m)}^{6} \cdot {(n)}^{0} + 6 \cdot {(m)}^{5} \cdot {(n)}^{1} + \dots + 1 \cdot {(m)}^{0} \cdot {(n)}^{6}$

Step 4

Simplify the polynomial result.

$m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3} + 15 m^{2} n^{4} + 6 m n^{5} + n^{6}$

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

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