Expand Using the Binomial Theorem (r+4)^3

${(r + 4)}^{3}$

Step 1

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

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(r)}^{3 - k} \cdot {(4)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

Multiply $r^{3}$ by $1$ .

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

Evaluate the exponent.

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

Multiply $4$ by $3$ .

$r^{3} + 12 r^{2} + 3 \cdot {(r)}^{1} \cdot {(4)}^{2} + 1 \cdot {(r)}^{0} \cdot {(4)}^{3}$

Simplify.

$r^{3} + 12 r^{2} + 3 \cdot r \cdot {(4)}^{2} + 1 \cdot {(r)}^{0} \cdot {(4)}^{3}$

Raise $4$ to the power of $2$ .

$r^{3} + 12 r^{2} + 3 r \cdot 16 + 1 \cdot {(r)}^{0} \cdot {(4)}^{3}$

Multiply $16$ by $3$ .

$r^{3} + 12 r^{2} + 48 r + 1 \cdot {(r)}^{0} \cdot {(4)}^{3}$

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

$r^{3} + 12 r^{2} + 48 r + {(r)}^{0} \cdot {(4)}^{3}$

Anything raised to $0$ is $1$ .

$r^{3} + 12 r^{2} + 48 r + 1 \cdot {(4)}^{3}$

Multiply ${(4)}^{3}$ by $1$ .

$r^{3} + 12 r^{2} + 48 r + {(4)}^{3}$

Raise $4$ to the power of $3$ .

$r^{3} + 12 r^{2} + 48 r + 64$

Expand Using the Binomial Theorem (r+4)^3

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