Expand Using the Binomial Theorem (m+n)^4

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

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(m)}^{4 - k} \cdot {(n)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} {(m)}^{4 - 0} \cdot {(n)}^{0} + \frac{4!}{(4 - 1)! 1!} {(m)}^{4 - 1} \cdot {(n)}^{1} + \frac{4!}{(4 - 2)! 2!} {(m)}^{4 - 2} \cdot {(n)}^{2} + \frac{4!}{(4 - 3)! 3!} {(m)}^{4 - 3} \cdot {(n)}^{3} + \frac{4!}{(4 - 4)! 4!} {(m)}^{4 - 4} \cdot {(n)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

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

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

Simplify.

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

Simplify.

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

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

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

Anything raised to $0$ is $1$ .

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

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

$m^{4} + 4 m^{3} n + 6 m^{2} n^{2} + 4 m n^{3} + n^{4}$

Expand Using the Binomial Theorem (m+n)^4

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