Expand Using the Binomial Theorem (m+1)^3

${(m + 1)}^{3}$

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

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(m)}^{3 - k} \cdot {(1)}^{k}$

Expand the summation.

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

Simplify the exponents for each term of the expansion.

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

Simplify each term.

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Multiply $1$ by ${(1)}^{0}$ by adding the exponents.

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Move ${(1)}^{0}$ .

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

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

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Raise $1$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $0$ and $1$ .

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

Simplify $1^{1} \cdot {(m)}^{3}$ .

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

Evaluate the exponent.

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

Multiply $3$ by $1$ .

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

Simplify.

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

One to any power is one.

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

Multiply $3$ by $1$ .

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

Multiply $1$ by ${(1)}^{3}$ by adding the exponents.

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Move ${(1)}^{3}$ .

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

Multiply ${(1)}^{3}$ by $1$ .

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Raise $1$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $3$ and $1$ .

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

Simplify $1^{4} \cdot {(m)}^{0}$ .

$m^{3} + 3 m^{2} + 3 m + 1^{4}$

One to any power is one.

$m^{3} + 3 m^{2} + 3 m + 1$

Expand Using the Binomial Theorem (m+1)^3