Expand Using the Binomial Theorem (r-3t)^4

${(r - 3 t)}^{4}$

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

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

Expand the summation.

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

Simplify the exponents for each term of the expansion.

$1 \cdot {(r)}^{4} \cdot {(- 3 t)}^{0} + 4 \cdot {(r)}^{3} \cdot (- 3 t) + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot (r) \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Simplify each term.

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

${(r)}^{4} \cdot {(- 3 t)}^{0} + 4 \cdot {(r)}^{3} \cdot {(- 3 t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Apply the product rule to $- 3 t$ .

$r^{4} \cdot ({(- 3)}^{0} t^{0}) + 4 \cdot {(r)}^{3} \cdot {(- 3 t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Rewrite using the commutative property of multiplication.

${(- 3)}^{0} r^{4} t^{0} + 4 \cdot {(r)}^{3} \cdot {(- 3 t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Anything raised to $0$ is $1$ .

$1 r^{4} t^{0} + 4 \cdot {(r)}^{3} \cdot {(- 3 t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

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

$r^{4} t^{0} + 4 \cdot {(r)}^{3} \cdot {(- 3 t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Anything raised to $0$ is $1$ .

$r^{4} \cdot 1 + 4 \cdot {(r)}^{3} \cdot {(- 3 t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

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

$r^{4} + 4 \cdot {(r)}^{3} \cdot {(- 3 t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Simplify.

$r^{4} + 4 r^{3} \cdot (- 3 t) + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Rewrite using the commutative property of multiplication.

$r^{4} + 4 \cdot - 3 (r^{3} t) + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Multiply $4$ by $- 3$ .

$r^{4} - 12 (r^{3} t) + 6 \cdot {(r)}^{2} \cdot {(- 3 t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Apply the product rule to $- 3 t$ .

$r^{4} - 12 r^{3} t + 6 r^{2} \cdot ({(- 3)}^{2} t^{2}) + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Rewrite using the commutative property of multiplication.

$r^{4} - 12 r^{3} t + 6 \cdot {(- 3)}^{2} (r^{2} t^{2}) + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Raise $- 3$ to the power of $2$ .

$r^{4} - 12 r^{3} t + 6 \cdot 9 (r^{2} t^{2}) + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Multiply $6$ by $9$ .

$r^{4} - 12 r^{3} t + 54 (r^{2} t^{2}) + 4 \cdot {(r)}^{1} \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Simplify.

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} + 4 \cdot r \cdot {(- 3 t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Apply the product rule to $- 3 t$ .

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} + 4 r \cdot ({(- 3)}^{3} t^{3}) + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Rewrite using the commutative property of multiplication.

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} + 4 \cdot {(- 3)}^{3} (r t^{3}) + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

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

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} + 4 \cdot - 27 (r t^{3}) + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

Multiply $4$ by $- 27$ .

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} - 108 (r t^{3}) + 1 \cdot {(r)}^{0} \cdot {(- 3 t)}^{4}$

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

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} - 108 r t^{3} + {(r)}^{0} \cdot {(- 3 t)}^{4}$

Anything raised to $0$ is $1$ .

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} - 108 r t^{3} + 1 \cdot {(- 3 t)}^{4}$

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

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} - 108 r t^{3} + {(- 3 t)}^{4}$

Apply the product rule to $- 3 t$ .

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} - 108 r t^{3} + {(- 3)}^{4} t^{4}$

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

$r^{4} - 12 r^{3} t + 54 r^{2} t^{2} - 108 r t^{3} + 81 t^{4}$

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