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

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $4 r$ .

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

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

$256 r^{4} \cdot {(t)}^{0} + 4 \cdot {(4 r)}^{3} \cdot {(t)}^{1} + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Anything raised to $0$ is $1$ .

$256 r^{4} \cdot 1 + 4 \cdot {(4 r)}^{3} \cdot {(t)}^{1} + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Multiply $256$ by $1$ .

$256 r^{4} + 4 \cdot {(4 r)}^{3} \cdot {(t)}^{1} + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Apply the product rule to $4 r$ .

$256 r^{4} + 4 \cdot (4^{3} r^{3}) \cdot {(t)}^{1} + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Multiply $4$ by $4^{3}$ by adding the exponents.

Move $4^{3}$ .

$256 r^{4} + 4^{3} \cdot 4 \cdot r^{3} \cdot {(t)}^{1} + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Multiply $4^{3}$ by $4$ .

Raise $4$ to the power of $1$ .

$256 r^{4} + 4^{3} \cdot 4^{1} \cdot r^{3} \cdot {(t)}^{1} + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

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

$256 r^{4} + 4^{3 + 1} \cdot r^{3} \cdot {(t)}^{1} + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Add $3$ and $1$ .

$256 r^{4} + 4^{4} \cdot r^{3} \cdot {(t)}^{1} + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Simplify $4^{4} \cdot r^{3} \cdot {(t)}^{1}$ .

$256 r^{4} + 4^{4} \cdot r^{3} \cdot t + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

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

$256 r^{4} + 256 \cdot r^{3} \cdot t + 6 \cdot {(4 r)}^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Apply the product rule to $4 r$ .

$256 r^{4} + 256 r^{3} t + 6 \cdot (4^{2} r^{2}) \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

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

$256 r^{4} + 256 r^{3} t + 6 \cdot (16 r^{2}) \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Multiply $16$ by $6$ .

$256 r^{4} + 256 r^{3} t + 96 r^{2} \cdot {(t)}^{2} + 4 \cdot {(4 r)}^{1} \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Simplify.

$256 r^{4} + 256 r^{3} t + 96 r^{2} t^{2} + 4 \cdot (4 r) \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Multiply $4$ by $4$ .

$256 r^{4} + 256 r^{3} t + 96 r^{2} t^{2} + 16 r \cdot {(t)}^{3} + 1 \cdot {(4 r)}^{0} \cdot {(t)}^{4}$

Multiply ${(4 r)}^{0}$ by $1$ .

$256 r^{4} + 256 r^{3} t + 96 r^{2} t^{2} + 16 r t^{3} + {(4 r)}^{0} \cdot {(t)}^{4}$

Apply the product rule to $4 r$ .

$256 r^{4} + 256 r^{3} t + 96 r^{2} t^{2} + 16 r t^{3} + 4^{0} r^{0} \cdot {(t)}^{4}$

Anything raised to $0$ is $1$ .

$256 r^{4} + 256 r^{3} t + 96 r^{2} t^{2} + 16 r t^{3} + 1 r^{0} \cdot {(t)}^{4}$

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

$256 r^{4} + 256 r^{3} t + 96 r^{2} t^{2} + 16 r t^{3} + r^{0} \cdot {(t)}^{4}$

Anything raised to $0$ is $1$ .

$256 r^{4} + 256 r^{3} t + 96 r^{2} t^{2} + 16 r t^{3} + 1 \cdot {(t)}^{4}$

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

$256 r^{4} + 256 r^{3} t + 96 r^{2} t^{2} + 16 r t^{3} + t^{4}$

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

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