Expand Using the Binomial Theorem (m+n)^7

${(m + n)}^{7}$

Step 1

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

$\sum_{k = 0}^{7} \frac{7!}{(7 - k)! k!} \cdot {(m)}^{7 - k} \cdot {(n)}^{k}$

Step 2

Expand the summation.

$\frac{7!}{(7 - 0)! 0!} {(m)}^{7 - 0} \cdot {(n)}^{0} + \frac{7!}{(7 - 1)! 1!} {(m)}^{7 - 1} \cdot {(n)}^{1} + \dots + \frac{7!}{(7 - 7)! 7!} {(m)}^{7 - 7} \cdot {(n)}^{7}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(m)}^{7} \cdot {(n)}^{0} + 7 \cdot {(m)}^{6} \cdot {(n)}^{1} + \dots + 1 \cdot {(m)}^{0} \cdot {(n)}^{7}$

Step 4

Simplify the polynomial result.

$m^{7} + 7 m^{6} n + 21 m^{5} n^{2} + 35 m^{4} n^{3} + 35 m^{3} n^{4} + 21 m^{2} n^{5} + 7 m n^{6} + n^{7}$

Expand Using the Binomial Theorem (m+n)^7

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